#6660330

order	512
nilpotency class	3
isoclinism	isoclinic class of #6636908
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1385 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 56 points:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=fgh,d^2=fi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,e^b=ei,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6567526);