C₂²×Q₈×#120 #72315145

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²¹

Aliases

C₂²×Q₈×#120

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂³×#120	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

302517 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 20 points:

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

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

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

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