#154185

order	512
nilpotency class	2
isoclinism	isoclinic class of #14715
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁵	C₂⁴
commutator	C₂⁴	C₂³×C₄
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

4614 conjugacy classes of subgroups; only the 818 normal subgroups are shown.

normal subgroups only

This group has 4614 conjugacy classes of subgroups — too many to draw. Only its 818 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=gh,b^a=be,c^a=cf,d^a=dh,c^b=cg

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, 61381);