C₂³×#2498 #72325458

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂³×#2498

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂³ ▷ C₁

121485 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 7 generators on 26 points:

(8,10)(11,12)(13,18)(14,17)(15,16)(19,20,23,25)(21,26,24,22);(8,10)(12,16)(13,17)(19,21,23,24)(20,22,25,26);(3,4);(5,6);(11,13)(12,14)(15,17)(16,18)(19,21)(20,22)(23,24)(25,26);(1,2)(3,4)(5,6);(7,8)(9,10)(11,14)(12,17)(13,16)(15,18)(19,22)(20,24)(21,25)(23,26)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#38364, C₂³×#35683, C₂³×#36737
maximal normal subgroup of	C₂³×#38364, C₂³×#35683, C₂³×#36737
maximal quotient of	C₂³×#14905

GAP

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

gap> G := Group((8,10)(11,12)(13,18)(14,17)(15,16)(19,20,23,25)(21,26,24,22), (8,10)(12,16)(13,17)(19,21,23,24)(20,22,25,26), (3,4), (5,6), (11,13)(12,14)(15,17)(16,18)(19,21)(20,22)(23,24)(25,26), (1,2)(3,4)(5,6), (7,8)(9,10)(11,14)(12,17)(13,16)(15,18)(19,22)(20,24)(21,25)(23,26));

C23×#2498 #72325458