#392440

order	512
nilpotency class	3
isoclinism	isoclinic class of #184786
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#95 ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2028 conjugacy classes of subgroups; only the 506 normal subgroups are shown.

normal subgroups only

This group has 2028 conjugacy classes of subgroups — too many to draw. Only its 506 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=f,b^a=be,c^a=cf,d^a=dg,g^a=gi,c^b=ci,h^d=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299636);