#6388741

order	512
nilpotency class	3
isoclinism	isoclinic class of #61591
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1778 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)(7,14)(8,12)(9,18)(10,13)(11,16)(15,20)(17,19)(21,22)(23,29)(24,26)(25,33)(27,31)(28,32)(30,36)(34,35);(1,3,4,2)(5,7)(6,11)(8,17)(9,15)(10,16)(12,20)(13,14)(18,19)(22,26)(27,35)(28,33)(29,36);(21,23,24,30)(22,27,26,35)(25,34,32,31)(28,36,33,29);(5,8,10,9)(6,12,13,18)(7,15,16,17)(11,19,14,20);(5,9,10,8)(6,12,13,18)(7,15,16,17)(11,20,14,19)(22,26)(27,35)(28,33)(29,36)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6295937);