#382795

order	512
nilpotency class	3
isoclinism	isoclinic class of #29592
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1538 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

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, 289991);