C₂³×#45599 #72301710

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁶

Aliases

direct product

C₂³×#45599

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

957192 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#7985294, C₂³×#8698894, C₂³×#8195808
maximal normal subgroup of	C₂³×#7985294, C₂³×#8698894, C₂³×#8195808

GAP

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

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

C23×#45599 #72301710