#6769667

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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	#38948	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1500 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fi,d^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^c=ei,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, 6676863);