#247034

order	512
nilpotency class	3
isoclinism	isoclinic class of #247017
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1039 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1039 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) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(2,6)(4,11)(7,15)(8,12)(10,13)(14,16)(17,18)(19,31)(20,36)(21,38)(22,29)(23,28)(24,30)(25,33)(26,37)(27,39)(32,44)(34,46)(35,45)(40,43)(41,48)(42,47);(3,9)(4,11)(6,12)(7,16)(17,19,24,35)(18,25,30,46)(20,21,32,40)(22,34,23,33)(26,38,47,43)(27,44,48,36)(28,45,29,31)(37,41,42,39);(1,2)(3,7)(4,12)(5,8)(6,11)(9,16)(10,14)(13,15)(17,20,24,32)(18,26,30,47)(19,21,35,40)(22,37,23,42)(25,43,46,38)(27,45,48,31)(28,36,29,44)(33,39,34,41);(1,3,5,9)(2,7,8,16)(4,13,11,10)(6,14,12,15)(17,21)(18,27)(19,32)(20,35)(22,41)(23,39)(24,40)(25,44)(26,45)(28,43)(29,38)(30,48)(31,47)(33,37)(34,42)(36,46)

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=i,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,g^a=gh,c^b=cg,d^b=di,e^b=eh,f^b=fhi,e^c=ei,f^c=fhi,e^d=eh,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 154230);