#493753

order	512
nilpotency class	4
isoclinism	isoclinic class of #480314
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
socle	C₂	C₂×#1799
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1103 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)(4,18,19,34)(5,22)(6,25,27,11)(7,12)(8,15)(9,21)(10,24)(14,38,39,61)(16,44,46,36)(17,37)(20,50,49,48)(23,33,55,53)(26,31)(28,57)(29,42,56,64)(30,43)(32,54)(35,41)(40,47,62,59)(45,60)(51,52)(58,63);(1,3,5,15,7,17,24,43)(2,8,10,13,12,30,22,37)(4,14,20,40,19,39,49,62)(6,16,23,42,27,46,55,64)(9,28,32,35,31,58,52,60)(11,29,33,36,25,56,53,44)(18,47,48,38,34,59,50,61)(21,41,51,57,26,45,54,63);(1,4)(2,9)(3,14)(5,20)(6,26)(7,19)(8,28)(10,32)(11,34)(12,31)(13,35)(15,40)(16,45)(17,39)(18,25)(21,27)(22,52)(23,54)(24,49)(29,59)(30,58)(33,50)(36,61)(37,60)(38,44)(41,46)(42,63)(43,62)(47,56)(48,53)(51,55)(57,64);(4,19)(9,31)(14,39)(18,34)(20,49)(21,26)(28,58)(32,52)(35,60)(38,61)(40,62)(41,45)(47,59)(48,50)(51,54)(57,63)

Polycyclic presentation:

2^9|a^2=g,b^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^c=di,f^c=fi,g^c=ghi,g^f=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303781, #72303803, #72303804, …
maximal normal subgroup of	#72303781, #72303803, #72303804, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400949);