#6833152

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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	C₂×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1274 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(2,5)(3,7)(9,10,12,13)(11,14,15,16)(17,18)(19,20)(21,27)(22,26)(23,28)(24,25)(29,32)(30,31);(1,2)(3,6)(4,5)(7,8)(17,19,22,25)(18,24,26,20)(21,32,30,23)(27,28,31,29);(10,13)(14,16)(17,20,23,31,22,24,32,27)(18,25,28,21,26,19,29,30);(1,3,4,7)(2,6,5,8)(18,26)(20,24)(27,31)(28,29);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,21,32,25,22,30,23,19)(18,27,29,24,26,31,28,20)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=h,d^2=fi,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6740348);