C₄×#2811 #521253

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

direct product

C₄×#2811

Subgroups

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

2578 conjugacy classes of subgroups; only the 1054 normal subgroups are shown.

normal subgroups only

This group has 2578 conjugacy classes of subgroups — too many to draw. Only its 1054 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) = 512

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

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

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, 428449);