C₂²×#14272 #72291894

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³×C₄³

Aliases

direct product

C₂²×#14272

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁴×#127	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

99061 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 24 points:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#687828, C₂²×#719145, C₂²×#719644, …
maximal normal subgroup of	C₂²×#687828, C₂²×#719145, C₂²×#719644, …
maximal quotient of	C₂×#72335141

GAP

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

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

C22×#14272 #72291894