#503137

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×D₄×D₈	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1392 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 24 points:

(2,3)(6,8)(10,14)(11,18)(12,19)(15,20)(16,24)(17,23)(21,22);(1,2,4,3)(10,14)(11,18)(12,19)(15,20)(16,24)(17,23)(21,22);(5,6,7,8)(9,10,15,24,12,16,11,14,13,17,18,22,19,21,20,23);(1,3,4,2)

Polycyclic presentation:

2^9|b^2=e,c^2=fhi,d^2=e,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310053, D₄×#2187, #72310196
maximal normal subgroup of	#72310053, D₄×#2187, #72310196

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410333);