C₂×#10211 #232818

order	512
nilpotency class	3
isoclinism	isoclinic class of #10208
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#10211

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1094 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 1094 conjugacy classes of subgroups — too many to draw. Only its 194 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140014);