#244568

order	512
nilpotency class	3
isoclinism	isoclinic class of #244547
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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 994 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,g^b=gi,e^c=ei,f^c=fh,g^c=gi,e^d=eh,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151764);