C₂²×#15984 #72295258

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁶

Aliases

direct product

C₂²×#15984

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁴×#121	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

85077 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:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293936, C₂²×#5021210, C₂×#72293958, …
maximal normal subgroup of	C₂×#72293936, C₂²×#5021210, C₂×#72293958, …
maximal quotient of	C₂×#72334395

GAP

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

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

C22×#15984 #72295258