#6388349

order	512
nilpotency class	3
isoclinism	isoclinic class of #61609
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#449	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1823 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=fi,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=dfi,g^c=gi,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6295545);