#153945

order	512
nilpotency class	2
isoclinism	isoclinic class of #2613
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

4622 conjugacy classes of subgroups; only the 882 normal subgroups are shown.

normal subgroups only

This group has 4622 conjugacy classes of subgroups — too many to draw. Only its 882 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 61141);