C₂³×#2815 #72296865

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁷×C₄²

Aliases

direct product

C₂³×#2815

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂³ ▷ C₁

155253 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#3332795, C₂³×#57971, C₂³×#40584, …
maximal normal subgroup of	C₂²×#3332795, C₂³×#57971, C₂³×#40584, …
maximal quotient of	C₂³×#16500, C₂³×#16252, C₂³×#16362

GAP

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

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

C23×#2815 #72296865