#473033

order	512
nilpotency class	3
isoclinism	isoclinic class of #454375
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

928 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)(13,14,16,17)(15,18,19,20)(21,22,26,30)(23,27,24,28)(25,29,31,34)(32,35,33,36);(1,2,4,3)(14,17)(15,19)(21,23,31,33,26,24,25,32)(22,27,34,36,30,28,29,35);(1,2)(3,4)(6,9)(10,12)(13,15)(14,18)(16,19)(17,20)(21,24,26,23)(22,28,30,27)(25,32,31,33)(29,35,34,36);(1,3,4,2)(5,7,8,11)(6,10,9,12)(15,19)(18,20)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 380229);