C₂×#7093 #122855

order	512
nilpotency class	4
isoclinism	isoclinic class of #7092
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂×#7093

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#331
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₈²	C₂³
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

274 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 512

A minimal faithful permutation representation — 3 generators on 34 points:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30051);