C₂²×#42091 #72301265

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁷×C₄

Aliases

direct product

C₂²×#42091

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

73697 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#9164052
maximal normal subgroup of	C₂²×#9164052

GAP

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

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

C22×#42091 #72301265