C₄×#2866 #607639

order	512
nilpotency class	2
isoclinism	isoclinic class of #2846
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₄×#2866

Subgroups

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

2768 conjugacy classes of subgroups; only the 978 normal subgroups are shown.

normal subgroups only

This group has 2768 conjugacy classes of subgroups — too many to draw. Only its 978 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)(7,11)(8,10)(9,12)(13,14)(15,18)(16,20)(17,19)(21,22)(23,25)(24,26)(27,28);(5,7,9,8)(6,10,12,11)(13,15,17,16)(14,18,19,20)(21,23,24,27)(22,25,26,28);(5,7,9,8)(6,11,12,10)(13,16,17,15)(14,18,19,20);(5,8,9,7)(6,10,12,11)(22,26)(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, 514835);