#6677422

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667700
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	#2408	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1999 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280286, #72280659, #72282227
maximal normal subgroup of	#72280286, #72280659, #72282227

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6584618);