#6537069

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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 ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1703 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(2,4)(22,26)(24,31)(27,32)(28,33)(30,35)(34,36);(1,2,3,4)(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)(22,27)(23,29)(26,32)(30,35);(6,10)(11,17)(12,18)(19,20)(21,22,25,27)(23,28,29,34)(24,32,31,26)(30,36,35,33);(5,7,9,14)(6,11,10,17)(8,15,16,13)(12,20,18,19)(21,23,25,29)(22,28,27,34)(24,30,31,35)(26,33,32,36);(5,8,9,16)(6,12,10,18)(7,13,14,15)(11,19,17,20)(23,29)(28,34)(30,35)(33,36)

Polycyclic presentation:

2^9|b^2=f,c^2=i,d^2=h,e^2=hi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6444265);