#244148

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

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

normal subgroups only

This group has 598 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:

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

Polycyclic presentation:

2^9|a^2=e,b^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, 151344);