#153848

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁵	C₂⁴
commutator	C₂³	C₂²×C₄²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

5282 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=gi,b^a=be,c^a=cf,d^a=de,c^b=cg

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 61044);