#7035257

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1231 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)(6,7)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26)(27,35)(28,32)(29,42)(30,33)(31,37)(34,44)(36,51)(38,50)(39,53)(40,49)(41,56)(43,46)(45,52)(47,54)(48,55);(1,2,4,5)(3,7,8,6)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,27,30,38)(26,31,33,46)(28,39,40,36)(29,43,44,37)(32,47,49,45)(34,50,42,35)(41,52,55,54)(48,51,56,53);(1,3,4,8)(2,6,5,7)(12,20)(15,24)(17,19)(21,23)(25,28,30,40)(26,32,33,49)(27,36,38,39)(29,41,44,55)(31,45,46,47)(34,48,42,56)(35,51,50,53)(37,52,43,54);(10,16)(11,18)(15,24)(17,19)(26,33)(28,40)(31,46)(34,42)(35,50)(36,39)(41,55)(52,54)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,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, 6942453);