#478148

order	512
nilpotency class	3
isoclinism	isoclinic class of #476082
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1034 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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=fi,c^2=fi,d^2=fi,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=def,d^c=de,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 385344);