C₂×#9977 #232437

order	512
nilpotency class	3
isoclinism	isoclinic class of #9974
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#9977

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3186	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1522 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1522 conjugacy classes of subgroups — too many to draw. Only its 210 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 — 4 generators on 34 points:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139633);