C₂²×#2239 #7623057

order	512
nilpotency class	4
isoclinism	isoclinic class of #2238
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂²×#2239

Subgroups

	subgroup	G/H
center, socle	C₂³	#481
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	C₂⁵	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1606 conjugacy classes of subgroups; only the 578 normal subgroups are shown.

normal subgroups only

This group has 1606 conjugacy classes of subgroups — too many to draw. Only its 578 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) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

(1,2);(3,4);(5,6,10,14)(7,15,16,33)(8,19,20,36)(9,24,22,13)(11,26,27,17)(12,29,30,23)(18,31,25,34)(21,28,35,32);(8,20)(12,30)(15,33)(17,26)(19,36)(21,35)(23,29)(31,34);(5,7,9,25,10,16,22,18)(6,11,13,32,14,27,24,28)(8,21,23,26,20,35,29,17)(12,31,19,33,30,34,36,15)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#33690, C₂²×#33691, C₂²×#33659
maximal normal subgroup of	C₂²×#33690, C₂²×#33691, C₂²×#33659

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530253);