C₂²×#2201 #7622860

order	512
nilpotency class	4
isoclinism	isoclinic class of #2199
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product

C₂²×#2201

Subgroups

	subgroup	G/H
center, socle	C₂³	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×D₄	C₂⁵
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂³×D₄ ◁ G
LECS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁

3826 conjugacy classes of subgroups; only the 578 normal subgroups are shown.

normal subgroups only

This group has 3826 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)(7,15)(8,12)(9,13)(10,14)(11,17)(16,32)(18,31)(19,30)(20,26)(21,27)(22,29)(23,33)(24,34)(25,35)(28,36);(5,7,10,18)(6,11,14,24)(8,19,21,16)(9,17,22,34)(12,25,27,23)(13,15,29,31)(20,35,36,33)(26,30,28,32);(5,8,10,21)(6,12,14,27)(7,16,18,19)(9,20,22,36)(11,23,24,25)(13,28,29,26)(15,30,31,32)(17,33,34,35)

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#33552
maximal normal subgroup of	C₂²×#33552
maximal quotient of	C₂²×#13349, C₂²×#13050

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530056);