C₂×#772851 #72299592

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

51751 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:

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

maximal subgroup of	C₂×#72298081, C₂×#72297414, C₂×#72261010
maximal normal subgroup of	C₂×#72298081, C₂×#72297414, C₂×#72261010

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

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