#389313

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

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 ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1264 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 296509);