C₂×#12464 #490553

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

Aliases

direct product

C₂×#12464

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

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, 397749);