#244183

order	512
nilpotency class	3
isoclinism	isoclinic class of #232783
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

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

normal subgroups only

This group has 668 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151379);