#244136

order	512
nilpotency class	3
isoclinism	isoclinic class of #232760
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	#2397	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

638 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=eh,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,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, 151332);