C₂×#152491 #72321881

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₆ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂×#152491

Subgroups

	subgroup	G/H
center	C₂×C₁₆	#120
commutator	C₂³	C₂³×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂⁴
socle	C₂²	#12976
prime radical	C₂×#525	C₈
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

1797 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 66
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 66 points:

(10,25)(26,45)(27,46)(28,47)(48,60)(49,61)(50,62)(63,66);(1,2);(3,4)(5,10)(6,12)(7,13)(8,14)(9,11)(15,26)(16,25)(17,27)(18,28)(19,32)(20,33)(21,29)(22,34)(23,30)(24,31)(35,45)(36,48)(37,49)(38,46)(39,47)(40,50)(41,54)(42,51)(43,52)(44,53)(55,60)(56,61)(57,63)(58,62)(59,64)(65,66);(3,5,6,15,7,17,19,36,8,18,20,37,22,40,41,57,9,16,21,35,23,38,42,55,24,39,43,56,44,58,59,65)(4,10,12,26,13,27,32,48,14,28,33,49,34,50,54,63,11,25,29,45,30,46,51,60,31,47,52,61,53,62,64,66)

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72217181, C₂×#72224993, C₂×#72225009, …
maximal normal subgroup of	C₂×#72217181, C₂×#72224993, C₂×#72225009, …

GAP

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

gap> G := Group((10,25)(26,45)(27,46)(28,47)(48,60)(49,61)(50,62)(63,66), (1,2), (3,4)(5,10)(6,12)(7,13)(8,14)(9,11)(15,26)(16,25)(17,27)(18,28)(19,32)(20,33)(21,29)(22,34)(23,30)(24,31)(35,45)(36,48)(37,49)(38,46)(39,47)(40,50)(41,54)(42,51)(43,52)(44,53)(55,60)(56,61)(57,63)(58,62)(59,64)(65,66), (3,5,6,15,7,17,19,36,8,18,20,37,22,40,41,57,9,16,21,35,23,38,42,55,24,39,43,56,44,58,59,65)(4,10,12,26,13,27,32,48,14,28,33,49,34,50,54,63,11,25,29,45,30,46,51,60,31,47,52,61,53,62,64,66));

C2×#152491 #72321881