C₂²×#2810 #7633603

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

direct product

C₂²×#2810

Subgroups

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

6002 conjugacy classes of subgroups; only the 3314 normal subgroups are shown.

normal subgroups only

This group has 6002 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) = 512

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

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

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, 7540799);