#244184

order	512
nilpotency class	3
isoclinism	isoclinic class of #232783
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

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

normal subgroups only

This group has 610 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:

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

Polycyclic presentation:

2^9|a^2=eh,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, 151380);