C₂×#62694 #10586117

order	512
nilpotency class	2
isoclinism	isoclinic class of #560
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂×#62694

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁷
socle	C₂³	C₂⁶
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂² ▷ C₁

32380 conjugacy classes of subgroups; only the 29250 normal subgroups are shown.

normal subgroups only

This group has 32380 conjugacy classes of subgroups — too many to draw. Only its 29250 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) = 34
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 7 generators on 34 points:

(1,2);(3,4)(5,11)(6,10)(7,9)(8,12)(13,17)(14,16)(15,18)(19,20)(21,24)(22,26)(23,25)(27,30)(28,31)(29,32)(33,34);(3,5)(4,8)(6,13)(7,12)(9,11)(10,15)(14,18)(16,17)(19,21)(20,24)(22,28)(23,27)(25,30)(26,31)(29,33)(32,34);(5,12)(8,11)(13,18)(15,17)(20,25)(21,27)(26,32)(28,33);(4,9)(8,11)(10,16)(15,17)(21,27)(24,30)(28,33)(31,34);(19,22,23,29)(20,26,25,32)(21,28,27,33)(24,31,30,34);(3,6,7,14)(4,10,9,16)(5,13,12,18)(8,15,11,17)

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×#11710658, #72261539, C₂×#72260447
maximal subgroup of	#72209967, C₂×#11710659, C₂×#62694⋊C₅, …
maximal normal subgroup of	#72209967, C₂×#11710659, C₂×#62694⋊C₅, …
maximal quotient of	C₂×#7633646, C₂×#7633500

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10493313);