#244130

order	512
nilpotency class	3
isoclinism	isoclinic class of #232766
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	#532	C₂³
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 142 normal subgroups are shown.

normal subgroups only

This group has 601 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151326);