C₂²×#12650 #72309779

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

direct product

C₂²×#12650

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#442
commutator	C₈	C₂³×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂³×#135	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂³×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

8469 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72256851, C₂²×#495721, C₂×#72249925, …
maximal normal subgroup of	C₂×#72256851, C₂²×#495721, C₂×#72249925, …
maximal quotient of	C₂²×#128688

GAP

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

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

C22×#12650 #72309779