C₂²×#14683 #72334495

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁶

Aliases

direct product

C₂²×#14683

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

82869 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)(10,12)(14,17)(15,19)(21,22)(23,24)(25,26)(27,28);(6,8)(9,10,11,12)(14,17)(15,19)(22,26)(24,28);(5,6)(7,8)(9,10)(11,12)(13,14,16,17)(15,18,19,20)(21,23,25,27)(22,24,26,28);(5,6)(7,8)(9,10)(11,12)(13,15)(14,18)(16,19)(17,20)(21,24)(22,27)(23,26)(25,28);(3,4);(1,2)(3,4)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#4308012, C₂²×#559859, C₂²×#613868, …
maximal normal subgroup of	C₂²×#4308012, C₂²×#559859, C₂²×#613868, …
maximal quotient of	C₂×#72334404

GAP

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

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

C22×#14683 #72334495