C₂×#7821 #165781

order	512
nilpotency class	3
isoclinism	isoclinic class of #7734
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#7821

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

1093 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1093 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);(4,8)(5,6)(9,10)(12,15)(13,16)(17,18)(19,20,23,26)(21,24,28,31)(22,25,29,32)(27,33,34,30);(3,4,7,9)(5,8,6,10)(11,12,14,15)(13,16,17,18)(19,21,23,28)(20,24,26,31)(22,27,29,34)(25,30,32,33);(3,5,7,6)(4,8,9,10)(11,13,14,17)(12,16,15,18)(19,22,23,29)(20,25,26,32)(21,27,28,34)(24,30,31,33)

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 72977);