C₂×#12463 #490552

order	512
nilpotency class	4
isoclinism	isoclinic class of #12463
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#12463

Subgroups

	subgroup	G/H
center, socle	C₂²	#1675
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#135	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

386 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 146 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);(3,4,10,16)(5,17,22,49)(6,23)(7,28,30,14)(8,13,32,44)(9,33,29,15)(11,34,38,19)(12,39)(18,50)(20,47,57,37)(21,36,55,48)(24,43)(25,59)(26,45)(27,40)(31,41)(35,52)(42,60)(46,61)(51,58)(53,63)(54,62)(56,65)(64,66);(3,5,8,20,9,21,30,34,10,22,32,57,29,55,7,19)(4,11,28,48,15,37,44,49,16,38,14,36,33,47,13,17)(6,18,31,56,25,52,39,61,27,54,60,66,26,53,24,51)(12,35,59,65,41,50,23,58,43,63,45,64,42,62,40,46);(3,6)(4,12)(5,18)(7,24)(8,31)(9,25)(10,27)(11,35)(13,40)(14,45)(15,41)(16,43)(17,46)(19,51)(20,56)(21,52)(22,54)(23,44)(26,29)(28,59)(30,39)(32,60)(33,42)(34,61)(36,64)(37,50)(38,63)(47,62)(48,65)(49,58)(53,55)(57,66)

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397748);