#6719832

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1235 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=fi,d^2=f,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6627028);