#96273

order	512
nilpotency class	3
isoclinism	isoclinic class of #7654
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

734 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=ghi,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, 3469);