C₂²×#2203 #7622893

order	512
nilpotency class	4
isoclinism	isoclinic class of #2202
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product

C₂²×#2203

Subgroups

	subgroup	G/H
center, socle	C₂³	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×Q₈	C₂⁵
prime radical	C₂²×#454	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

3702 conjugacy classes of subgroups; only the 578 normal subgroups are shown.

normal subgroups only

This group has 3702 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)(7,16)(8,20)(9,12)(10,15)(11,17)(13,23)(14,25)(18,33)(19,31)(21,29)(22,30)(24,34)(26,32)(27,35)(28,36);(5,7,10,19)(6,11,15,29)(8,21,23,17)(9,24,25,18)(12,27,14,28)(13,31,20,16)(22,35,26,36)(30,33,32,34);(6,12)(9,25)(11,27)(13,32)(14,15)(16,33)(18,24)(20,30)(22,26)(28,29)(31,34)(35,36)

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267550
maximal normal subgroup of	#72267550
maximal quotient of	C₂²×#13379

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530089);