C₂²×#2736 #7633530

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁰×C₄²

Aliases

direct product

C₂²×#2736

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁶
prime radical	C₂⁷	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

14162 conjugacy classes of subgroups; only the 3314 normal subgroups are shown.

normal subgroups only

This group has 14162 conjugacy classes of subgroups — too many to draw. Only its 3314 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 — 6 generators on 28 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂²×#2736⋊C₃
maximal subgroup of	#72219198, #72219205, #72267508, …
maximal normal subgroup of	#72219198, #72219205, #72267508, …
maximal quotient of	#72219203, #72240904, #72267502, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7540726);