#6653999

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632074
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	#37379	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1586 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=f,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efhi,g^a=gi,c^b=ch,d^b=dfh,g^c=gi,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, 6561195);