C₂×#72233677 #72324045

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²²

Aliases

	subgroup	G/H
center, socle	C₂⁵	C₂⁶
commutator, Frattini, cosocle	C₂⁴	C₂⁷
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 9776 conjugacy classes of subgroups; only the 45880 normal subgroups are shown.

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

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

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

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

maximal subgroup of	C₂×D₄²×#120, C₂×#72334382
maximal normal subgroup of	C₂×D₄²×#120, C₂×#72334382

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

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