#440833

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

1232 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=egi,c^2=e,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,f^c=fi,h^c=hi,f^d=fi,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, 348029);