C₂×#845351 #72299327

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

50605 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:

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

maximal subgroup of	C₂×#72298096, C₂×#72298083, C₂×#72297414
maximal normal subgroup of	C₂×#72298096, C₂×#72298083, C₂×#72297414

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

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