C₂×#8515 #187731

order	512
nilpotency class	3
isoclinism	isoclinic class of #8510
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂×#8515

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3567	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2064 conjugacy classes of subgroups; only the 314 normal subgroups are shown.

normal subgroups only

This group has 2064 conjugacy classes of subgroups — too many to draw. Only its 314 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)(5,13,16,9)(6,17)(7,20)(8,11)(10,19)(12,23)(14,30,31,34)(15,25,21,27)(18,28)(22,26)(24,32,29,33)(35,36,40,45,41,46,39,44)(37,47,51,42,50,57,49,56)(38,43,52,58,53,59,54,60)(48,62,63,66,64,61,65,55);(3,5)(4,9)(6,14)(7,21)(8,16)(10,24)(11,13)(12,27)(15,23)(17,30)(18,31)(19,33)(20,25)(22,32)(26,29)(28,34)(35,37)(36,42)(38,48)(39,51)(40,49)(41,50)(43,55)(44,57)(45,47)(46,56)(52,65)(53,64)(54,63)(58,61)(59,66)(60,62);(3,6,8,18)(4,10,11,26)(5,14,16,31)(7,22,23,19)(9,24,13,29)(12,28,20,17)(15,33,21,32)(25,30,27,34)(35,38)(36,43)(37,48)(39,54)(40,52)(41,53)(42,55)(44,60)(45,58)(46,59)(47,61)(49,65)(50,64)(51,63)(56,66)(57,62)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 94927);