C₂⁴×#2849 #72310503

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁸

Aliases

direct product

C₂⁴×#2849

Subgroups

	subgroup	G/H
center, socle	C₂⁷	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁸
prime radical	C₂⁵×#120	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁷ ◁ G
LECS	G ▷ C₂³ ▷ C₁

9507032 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 8 generators on 24 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂⁴×#36614
maximal normal subgroup of	C₂⁴×#36614

GAP

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

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

C24×#2849 #72310503