#244116

order	512
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2397
prime radical	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 607 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151312);