C₂×#674693 #72299404

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴×C₄

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

52693 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 512

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

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

maximal subgroup of	C₂×#72297594, C₂×#72297397, C₂×#72261412
maximal normal subgroup of	C₂×#72297594, C₂×#72297397, C₂×#72261412

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

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