#244167

order	512
nilpotency class	3
isoclinism	isoclinic class of #232789
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	#3100	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

604 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,d^2=ef,e^2=h,f^2=i,b^a=be,c^a=cf,g^a=ghi,c^b=cg,d^b=di,e^b=eh,f^b=fh,d^c=dh,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, 151363);