#7010374

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1277 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,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, 6917570);