#244132

order	512
nilpotency class	3
isoclinism	isoclinic class of #232760
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	C₂²×D₄	#41
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

590 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:

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

Polycyclic presentation:

2^9|a^2=e,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=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, 151328);