C₂²×#7630442 #72339332

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁸

Aliases

direct product

C₂²×#7630442

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁸
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5639608 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 8 generators on 28 points:

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

Conjugacy classes & character table

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

Relationships

maximal quotient of

C₂×#72339320, C₂²×#72233780

GAP

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

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

C22×#7630442 #72339332