C₂×#7625795 #72298866

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₄ · Schur multiplier (H₂) = C₂²⁰

Aliases

direct product

C₂×#7625795

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator	C₂²	C₂⁶×C₄
Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂²×#3484	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

129525 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 7 generators on 18 points:

(8,10)(12,14)(16,17);(3,4,6,5)(7,8)(9,10)(11,12,15,14)(13,16,18,17);(3,4,6,5)(12,14)(16,17);(3,4,6,5)(11,13,15,18)(12,16,14,17);(3,4,6,5)(11,13)(12,16)(14,17)(15,18);(3,5,6,4)(11,14,15,12)(13,17,18,16);(1,2)

Conjugacy classes & character table

340 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×#72232066, #72342380
maximal normal subgroup of	C₂×#72232066, #72342380
maximal quotient of	C₂×#72297598, C₂×#72298087, C₂×#72232121, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((8,10)(12,14)(16,17), (3,4,6,5)(7,8)(9,10)(11,12,15,14)(13,16,18,17), (3,4,6,5)(12,14)(16,17), (3,4,6,5)(11,13,15,18)(12,16,14,17), (3,4,6,5)(11,13)(12,16)(14,17)(15,18), (3,5,6,4)(11,14,15,12)(13,17,18,16), (1,2));

C2×#7625795 #72298866