#363589

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
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

1290 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)(7,9)(8,10)(11,12)(21,22)(23,28)(24,29)(25,27)(26,30)(31,36)(32,35)(33,34);(5,7,8,11)(6,9,10,12)(13,14)(15,18)(16,17)(19,20)(22,27)(23,31)(24,25)(28,35)(29,30)(34,36);(6,10)(9,12)(14,17)(18,20)(21,23,26,31)(22,28,30,36)(24,32,25,33)(27,34,29,35);(13,15,16,19)(14,18,17,20)(21,24)(23,32)(25,26)(27,29)(31,33)(34,35)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=i,d^2=gi,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,e^b=ei,e^d=ei

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, 270785);