#96266

order	512
nilpotency class	3
isoclinism	isoclinic class of #7654
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

1182 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=g,f^2=i,b^a=bd,c^a=ci,d^a=dg,f^a=fgh,d^b=dh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3462);