C₂²×#1780 #6370680

order	512
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹²

Aliases

direct product

C₂²×#1780

Subgroups

	subgroup	G/H
center	C₂⁴×C₄	C₂³
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂⁵	C₂²×C₄
prime radical	C₂⁴×#41	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

18906 conjugacy classes of subgroups; only the 2114 normal subgroups are shown.

normal subgroups only

This group has 18906 conjugacy classes of subgroups — too many to draw. Only its 2114 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) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

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, 6277876);