C₂²×#2263 #7625188

order	512
nilpotency class	5
isoclinism	isoclinic class of #2263
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂²×#2263

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×D₁₆
commutator, Frattini, cosocle	C₁₆	C₂⁵
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

1474 conjugacy classes of subgroups; only the 538 normal subgroups are shown.

normal subgroups only

This group has 1474 conjugacy classes of subgroups — too many to draw. Only its 538 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)(7,11)(8,20)(9,23)(10,12)(13,16)(14,17)(15,21)(18,30)(19,24)(22,36)(25,32)(26,31)(27,33)(28,34)(29,35);(6,11)(7,16)(8,17)(9,21)(12,24)(13,14)(15,30)(18,35)(19,33)(20,25)(22,34)(23,26)(27,28)(29,36)(31,32);(6,12)(11,24)(13,27)(14,28)(15,23)(20,36)(25,29)(26,30)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	A₄×#2263, S₄×#2263, #72342900
maximal subgroup of	C₂²×#13658, C₂×C₄×#2263, C₂×D₃×#2263, …
maximal normal subgroup of	C₂²×#13658, C₂×C₄×#2263, C₂×#152549, …
maximal quotient of	C₂×C₄×#2263, C₂×#513082, #72234321, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532384);