#244189

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	#10012	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

635 conjugacy classes of subgroups.

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)(3,11,13,16,14,15,12,7)(17,18,21,25,22,26,33,40)(19,27,29,36,30,42,35,23)(20,24,34,41,32,39,47,48)(28,43,44,38,31,45,46,37);(1,3)(2,7)(4,12)(5,13)(6,14)(8,15)(9,11)(10,16)(17,19)(18,23)(20,31)(21,35)(22,30)(24,38)(25,42)(26,36)(27,40)(28,32)(29,33)(34,44)(37,39)(41,43)(45,48)(46,47);(2,8)(4,5)(7,15)(9,10)(11,16)(12,13)(20,32)(23,36)(24,39)(27,42)(28,31)(34,47)(41,48)(44,46);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,15,16,11)(17,20,33,47,22,32,21,34)(18,24,40,48,26,39,25,41)(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=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, 151385);