#7167144

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993899
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60835	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1655 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,f^d=fi

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, 7074340);