#244180

order	512
nilpotency class	3
isoclinism	isoclinic class of #232809
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

598 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)(3,11,12,16,14,15,13,7)(17,18,21,25,22,26,33,39)(19,27,29,36,30,41,35,23)(20,31,34,47,32,24,38,40)(28,43,45,37,44,42,48,46);(1,3,6,14)(2,7,10,16)(4,13,5,12)(8,15,9,11)(17,19,22,30)(18,23,26,36)(20,28,32,44)(21,35,33,29)(24,37,31,46)(25,41,39,27)(34,48,38,45)(40,43,47,42);(2,8)(4,5)(7,15)(9,10)(11,16)(12,13)(20,32)(23,36)(24,31)(27,41)(28,44)(34,38)(40,47)(45,48);(1,4,6,5)(2,9,10,8)(3,12,14,13)(7,15,16,11)(17,20,33,38,22,32,21,34)(18,24,39,47,26,31,25,40)(19,28,29,45,30,44,35,48)(23,37,27,42,36,46,41,43)

Polycyclic presentation:

2^9|a^2=e,b^2=h,d^2=efh,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dh,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

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151376);