C₂×#7098 #122873

order	512
nilpotency class	4
isoclinism	isoclinic class of #7092
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#7098

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

306 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 114 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);(3,4,5,10,6,9,13,17,7,11,14,18,15,8,12,16)(20,24)(21,27)(22,29)(25,34)(26,33)(28,31)(30,32);(4,8)(5,12)(6,15)(9,11)(10,18)(13,14)(19,20,23,26)(21,28,30,34)(22,31,29,25)(24,32,33,27)

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, 30069);