C₂×#7085 #122615

order	512
nilpotency class	4
isoclinism	isoclinic class of #7076
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#7085

Subgroups

	subgroup	G/H
center, socle	C₂³	#327
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₂²⋊C₈	C₂³
prime radical	C₂⁵	SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

460 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 128

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

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29811);