#473389

order	512
nilpotency class	3
isoclinism	isoclinic class of #454382
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 ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

902 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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=fgh,c^2=i,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de,g^c=gi

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, 380585);