#422934

order	512
nilpotency class	3
isoclinism	isoclinic class of #422933
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15310
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1863 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,e^c=ei,g^c=gi,e^d=ei,g^d=gi,h^d=hi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285398, #72286090, #72286213
maximal normal subgroup of	#72285398, #72286090, #72286213

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 330130);