C₂²×#9072 #72320922

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂¹²

Aliases

direct product

C₂²×#9072

Subgroups

	subgroup	G/H
center	C₂⁴×C₄	C₂×D₄
commutator	C₂×C₄	C₂⁴×C₈
Frattini, cosocle	C₂×C₄²	C₂⁵
socle	C₂⁵	C₂×C₂²⋊C₄
prime radical	C₂³×#121	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄ ◁ C₂⁴×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

18669 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 24 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#210728
maximal normal subgroup of	C₂²×#210728

GAP

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

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

C22×#9072 #72320922