#267872

order	512
nilpotency class	3
isoclinism	isoclinic class of #267851
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

869 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:

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

Polycyclic presentation:

2^9|a^2=fh,c^2=i,d^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^b=fh,g^b=gh,e^c=eh,f^c=fi,g^c=gh,f^d=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 175068);