C₂×#7824 #165733

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

Aliases

direct product

C₂×#7824

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

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, 72929);