C₂×#7524 #153482

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

Aliases

direct product

C₂×#7524

Subgroups

	subgroup	G/H
center, socle	C₂²	#1404
commutator	#125	C₂²×C₄
Frattini, cosocle	#490	C₂³
prime radical	C₂×#127	D₄
LCS	G ▷ #125 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#490 ◁ G
LECS	G ▷ #490 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 301 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)(5,13)(6,17,19,10)(7,12)(8,11,21,29)(9,14,24,32)(15,31)(16,30,22,34)(18,25,33,26)(20,27,23,28);(3,5,7,15)(4,9,12,26)(6,18,20,14)(8,22,21,16)(10,13,28,31)(11,25,29,24)(17,34,27,30)(19,32,23,33)

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, 60678);