C₂²×#2642 #7627786

order	512
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

direct product

C₂²×#2642

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

8402 conjugacy classes of subgroups; only the 3746 normal subgroups are shown.

normal subgroups only

This group has 8402 conjugacy classes of subgroups — too many to draw. Only its 3746 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) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 32 points:

(1,2);(3,4);(5,6,7,8)(11,15)(13,16)(18,22)(23,25)(24,30)(31,32);(5,6,7,8)(9,10,12,14)(11,13,15,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,32)(28,31)(29,30);(5,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,21,27)(18,23,22,25)(20,28,29,26)(24,32,30,31);(17,20,21,29)(18,24,22,30)(19,26,27,28)(23,31,25,32)

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, 7534982);