#243850

order	512
nilpotency class	3
isoclinism	isoclinic class of #232755
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#399
prime radical	C₂×#121	Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

663 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 663 conjugacy classes of subgroups — too many to draw. Only its 180 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151046);