C₂×#612457 #72323908

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵

Aliases

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

53021 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 18 points:

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

maximal subgroup of	C₂×#72323245, C₂×#72323269, C₂×#72261392
maximal normal subgroup of	C₂×#72323245, C₂×#72323269, C₂×#72261392

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

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