#6855691

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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 ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1193 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6762887);