C₂×#7525 #153487

order	512
nilpotency class	6
isoclinism	isoclinic class of #7523
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#7525

Subgroups

	subgroup	G/H
center, socle	C₂²	#1404
commutator	#125	C₂²×C₄
Frattini, cosocle	#491	C₂³
prime radical	C₂×#562	C₄
LCS	G ▷ #125 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#491 ◁ G
LECS	G ▷ #491 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

314 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 50 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 — 3 generators on 34 points:

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

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60683);