#7010628

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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

1268 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)(17,18,22,27,23,28,21,36)(19,24,31,42,32,43,37,41)(20,33,34,45,35,26,38,25)(29,40,47,44,30,48,46,39);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,19,23,32)(18,24,28,43)(20,30,35,29)(21,37,22,31)(25,44,45,39)(26,40,33,48)(27,42,36,41)(34,46,38,47);(9,11)(10,13)(12,16)(14,15)(17,20,23,35)(18,25,28,45)(19,29,32,30)(21,34,22,38)(24,39,43,44)(26,36,33,27)(31,46,37,47)(40,42,48,41);(2,5)(6,8)(11,16)(13,15)(17,20)(18,26)(19,30)(21,38)(22,34)(23,35)(24,40)(25,27)(28,33)(29,32)(31,46)(36,45)(37,47)(39,41)(42,44)(43,48);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,23,22)(18,27,28,36)(19,31,32,37)(20,34,35,38)(24,41,43,42)(25,26,45,33)(29,46,30,47)(39,40,44,48)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=di,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

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6917824);