#7622452

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622451
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂³×#99
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂×D₄	C₂⁵
prime radical	#62848	C₂
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×Q₈ ◁ C₂²×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁

4079 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 64 points:

(1,2)(3,13)(4,9)(5,10)(6,24,25,11)(7,12)(8,17,28,44)(14,35)(15,36)(16,37)(18,29)(19,46,47,30)(20,31)(21,32)(22,49,50,33)(23,34)(26,40,52,62)(27,43,54,64)(38,57)(39,58)(41,59)(42,60)(45,56,48,55)(51,61,53,63);(1,3)(2,8)(4,14)(5,15)(6,17)(7,16)(9,26)(10,27)(11,13)(12,28)(18,38)(19,40)(20,39)(21,41)(22,43)(23,42)(24,37)(25,44)(29,51)(30,35)(31,52)(32,53)(33,36)(34,54)(45,61)(46,58)(47,62)(48,63)(49,60)(50,64)(55,57)(56,59);(1,4,7,20)(2,9,12,31)(3,14,16,39)(5,21,23,18)(6,19,25,47)(8,26,28,52)(10,32,34,29)(11,30,24,46)(13,35,37,58)(15,41,42,38)(17,40,44,62)(22,48,50,45)(27,53,54,51)(33,56,49,55)(36,59,60,57)(43,63,64,61);(1,5,7,23)(2,10,12,34)(3,15,16,42)(4,18,20,21)(6,22,25,50)(8,27,28,54)(9,29,31,32)(11,33,24,49)(13,36,37,60)(14,38,39,41)(17,43,44,64)(19,45,47,48)(26,51,52,53)(30,55,46,56)(35,57,58,59)(40,61,62,63);(3,16)(8,28)(13,37)(14,39)(15,42)(17,44)(26,52)(27,54)(35,58)(36,60)(38,41)(40,62)(43,64)(51,53)(57,59)(61,63)

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=i,b^a=bf,f^a=fh,h^a=hi,e^b=ei,g^b=gh,d^c=di,g^f=gi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72302929, #72302953, #72303038, …
maximal normal subgroup of	#72302929, #72302953, #72303038, …
maximal quotient of	#72312403, #72312825

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529648);