C₂×#10531 #238984

order	512
nilpotency class	3
isoclinism	isoclinic class of #10530
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₂×#10531

Subgroups

	subgroup	G/H
center, socle	C₂³	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3253	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1010 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1010 conjugacy classes of subgroups — too many to draw. Only its 202 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) = 256

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

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 146180);