C₂²×#38502 #72326112

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸

Aliases

direct product

C₂²×#38502

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂²×#2433	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

79189 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 28 points:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#8598997
maximal normal subgroup of	C₂²×#8598997
maximal quotient of	C₂²×#525856

GAP

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

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

C22×#38502 #72326112