C₂²×#2658 #7627788

order	512
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂²×#2658

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁶
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

14010 conjugacy classes of subgroups; only the 4258 normal subgroups are shown.

normal subgroups only

This group has 14010 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) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 28 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534984);