#362120

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	D₄×C₂²⋊C₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1286 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 269316);