C₂³×#40588 #72300226

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁵×C₄

Aliases

direct product

C₂³×#40588

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁸
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂³ ▷ C₁

1764320 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 8 generators on 26 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#7849701
maximal normal subgroup of	C₂³×#7849701

GAP

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

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

C23×#40588 #72300226