#244135

order	512
nilpotency class	3
isoclinism	isoclinic class of #232760
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

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

normal subgroups only

This group has 696 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:

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

Polycyclic presentation:

2^9|a^2=e,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, 151331);