C₂⁴×#2684 #72325777

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁶×C₄

Aliases

direct product

C₂⁴×#2684

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

3637928 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 8 generators on 28 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#1262353
maximal normal subgroup of	C₂³×#1262353

GAP

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

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

C24×#2684 #72325777