C₂×#35456 #7628824

order	512
nilpotency class	2
isoclinism	isoclinic class of #35346
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

direct product

C₂×#35456

Subgroups

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

6450 conjugacy classes of subgroups; only the 3234 normal subgroups are shown.

normal subgroups only

This group has 6450 conjugacy classes of subgroups — too many to draw. Only its 3234 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) = 34
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 34 points:

(1,2);(5,9)(7,10)(12,15)(13,17)(19,20,24,25)(21,27,28,22)(23,26,31,32)(29,33,34,30);(3,4,6,8)(5,7,9,10)(11,12)(13,16)(14,15)(17,18)(20,25)(21,28)(23,31)(30,33);(3,5)(4,7)(6,9)(8,10)(11,13,14,17)(12,16,15,18)(19,21,24,28)(20,22,25,27)(23,29,31,34)(26,30,32,33);(19,22)(20,21)(23,30)(24,27)(25,28)(26,29)(31,33)(32,34);(19,23)(20,26)(21,29)(22,30)(24,31)(25,32)(27,33)(28,34)

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, 7536020);