C₂²×#13213 #72313515

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

12325 conjugacy classes of subgroups.

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 24 points:

(6,8)(10,14)(11,18)(12,19)(15,17)(16,23)(20,24)(21,22);(1,2);(3,4);(5,6,7,8);(9,10,11,15,19,21,24,22,13,17,20,14,12,16,18,23)

maximal subgroup of	C₂²×#498331, C₂²×#496124, C₂²×D₄×#361
maximal normal subgroup of	C₂²×#498331, C₂²×#496124, C₂²×D₄×#361

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

gap> G := Group((6,8)(10,14)(11,18)(12,19)(15,17)(16,23)(20,24)(21,22), (1,2), (3,4), (5,6,7,8), (9,10,11,15,19,21,24,22,13,17,20,14,12,16,18,23));