#123423

order	512
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂⁴×C₄	C₂³
commutator	C₂³	C₄³
socle, prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

2390 conjugacy classes of subgroups; only the 914 normal subgroups are shown.

normal subgroups only

This group has 2390 conjugacy classes of subgroups — too many to draw. Only its 914 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) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2)(3,4)(5,6)(7,8)(17,18,20,23,21,24,27,30)(19,22,26,29,25,28,31,32);(2,4)(9,10,11,12)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32);(6,8)(13,14,15,16)(19,25)(22,28)(26,31)(29,32)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,f^2=i,b^a=bd,c^a=ce,c^b=ci

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30619);