C₂²×#13924 #72339132

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁵×C₄

Aliases

direct product

C₂²×#13924

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁷	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

75141 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 28 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#521530, C₂²×#637046, C₂²×#637036, …
maximal normal subgroup of	C₂²×#521530, C₂²×#637046, C₂²×#637036, …
maximal quotient of	C₂×#72337369

GAP

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

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

C22×#13924 #72339132