C₂×#8736 #190016

order	512
nilpotency class	3
isoclinism	isoclinic class of #8736
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#8736

Subgroups

	subgroup	G/H
center, socle	C₂³	#381
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁶	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1414 conjugacy classes of subgroups; only the 282 normal subgroups are shown.

normal subgroups only

This group has 1414 conjugacy classes of subgroups — too many to draw. Only its 282 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) = 64

A minimal faithful permutation representation — 4 generators on 34 points:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 97212);