C₂⁴×#3532 #72295524

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁹ · Schur multiplier (H₂) = C₂³⁴

Aliases

direct product

C₂⁴×#3532

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁹
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂² ▷ C₁

11981096 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 9 generators on 24 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂⁴×#63011, C₂⁴×#62817, C₂³×#7792761
maximal normal subgroup of	C₂⁴×#63011, C₂⁴×#62817, C₂³×#7792761
maximal quotient of	C₂³×#8698894

GAP

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

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

C24×#3532 #72295524