#7125166

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993875
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	#61187	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1665 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=h,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,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, 7032362);