C₂²×#2657 #7627779

order	512
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄³

Aliases

direct product

C₂²×#2657

Subgroups

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

20746 conjugacy classes of subgroups; only the 4258 normal subgroups are shown.

normal subgroups only

This group has 20746 conjugacy classes of subgroups — too many to draw. Only its 4258 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) = 20
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 20 points:

(1,2);(3,4);(5,6,7,8)(13,14)(15,17)(16,18)(19,20);(9,10,11,12)(13,14,16,18)(15,17,20,19);(5,6)(7,8)(9,10)(11,12)(13,14)(15,19)(16,18)(17,20);(13,15,16,20)(14,17,18,19)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72299662
maximal subgroup of	C₂²×#37674, C₂²×#35921, C₂²×#40490, …
maximal normal subgroup of	C₂²×#37674, C₂²×#35921, C₂²×#40490, …
maximal quotient of	#72261535, C₂²×#16502, C₂²×#15142, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534975);