C₂×#8130 #181418

order	512
nilpotency class	3
isoclinism	isoclinic class of #8101
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

direct product

C₂×#8130

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#95
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁵	C₄²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1460 conjugacy classes of subgroups; only the 594 normal subgroups are shown.

normal subgroups only

This group has 1460 conjugacy classes of subgroups — too many to draw. Only its 594 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 — 4 generators on 34 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88614);