#478142

order	512
nilpotency class	3
isoclinism	isoclinic class of #476082
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

958 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^9|a^2=efi,b^2=f,c^2=f,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, 385338);