#450253

order	512
nilpotency class	3
isoclinism	isoclinic class of #417442
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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

1148 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:

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

Polycyclic presentation:

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

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, 357449);