C₂×#32877 #6631465

order	512
nilpotency class	3
isoclinism	isoclinic class of #32875
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂×#32877

Subgroups

	subgroup	G/H
center, socle	C₂²	#2842
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1776 conjugacy classes of subgroups; only the 490 normal subgroups are shown.

normal subgroups only

This group has 1776 conjugacy classes of subgroups — too many to draw. Only its 490 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 — 5 generators on 34 points:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6538661);