#244151

order	512
nilpotency class	3
isoclinism	isoclinic class of #232771
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	#2397
prime radical	C₂²×D₄	#41
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

601 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 601 conjugacy classes of subgroups — too many to draw. Only its 150 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:

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

Polycyclic presentation:

2^9|a^2=eh,c^2=h,d^2=efh,e^2=h,f^2=i,b^a=be,c^a=cf,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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151347);