#6910213

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#37683	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1826 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)(3,7)(17,18,21,23)(19,27,28,41)(20,32,33,25)(22,35,29,26)(24,37,36,31)(30,44,42,40)(34,39,43,47)(38,45,46,48);(1,2,4,5)(3,6,7,8)(9,10)(11,15)(12,14)(13,16)(18,23)(19,28)(25,32)(26,35)(30,42)(31,37)(39,47)(45,48);(3,7)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,29,37,21,28,22,31)(18,24,35,41,23,36,26,27)(20,30,43,48,33,42,34,45)(25,38,47,44,32,46,39,40);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(17,20,21,33)(18,25,23,32)(19,30,28,42)(22,34,29,43)(24,38,36,46)(26,39,35,47)(27,40,41,44)(31,45,37,48);(2,5)(6,8)(10,14)(13,15)(20,33)(25,32)(30,42)(34,43)(38,46)(39,47)(40,44)(45,48)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6817409);