#244157

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	#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

595 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 595 conjugacy classes of subgroups — too many to draw. Only its 142 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:

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

Polycyclic presentation:

2^9|a^2=eh,c^2=h,d^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, 151353);