#497332

order	512
nilpotency class	4
isoclinism	isoclinic class of #497329
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	D₄×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#61061	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#578 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1093 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,e^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,c^b=ci,d^b=de,e^b=ei,g^c=gh,e^d=ei,g^f=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342045
maximal normal subgroup of	#72342045

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404528);