C₂³×#57971 #72297019

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁴×C₄

Aliases

direct product

C₂³×#57971

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

1383720 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 8 generators on 30 points:

(16,19)(20,22)(23,24)(25,29)(26,27)(28,30);(9,13)(12,14)(24,27)(28,29);(8,11)(9,13)(16,19)(17,21)(25,30)(28,29);(7,8)(9,14)(10,11)(12,13)(15,16)(17,20)(18,19)(21,22)(23,25)(24,28)(26,30)(27,29);(7,9)(8,12)(10,13)(11,14)(15,17)(16,20)(18,21)(19,22)(23,24)(25,29)(26,27)(28,30);(3,4);(5,6);(1,2)(3,4)(5,6)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#72231349
maximal normal subgroup of	C₂²×#72231349

GAP

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

gap> G := Group((16,19)(20,22)(23,24)(25,29)(26,27)(28,30), (9,13)(12,14)(24,27)(28,29), (8,11)(9,13)(16,19)(17,21)(25,30)(28,29), (7,8)(9,14)(10,11)(12,13)(15,16)(17,20)(18,19)(21,22)(23,25)(24,28)(26,30)(27,29), (7,9)(8,12)(10,13)(11,14)(15,17)(16,20)(18,21)(19,22)(23,24)(25,29)(26,27)(28,30), (3,4), (5,6), (1,2)(3,4)(5,6));

C23×#57971 #72297019