#360895

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
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

1304 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 268091);