C₄×#2840 #544340

order	512
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄²

Aliases

direct product

C₄×#2840

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator	C₂³	C₂⁴×C₄
Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3566 conjugacy classes of subgroups; only the 882 normal subgroups are shown.

normal subgroups only

This group has 3566 conjugacy classes of subgroups — too many to draw. Only its 882 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) = 128

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

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	Q₈×#2840
maximal normal subgroup of	Q₈×#2840
maximal quotient of	#72267612

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 451536);