#7124124

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	#61276	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1579 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)(3,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,18,22,26)(19,28,33,41)(20,25,36,43)(21,37,38,27)(23,32,40,31)(24,34,42,35)(29,46,48,39)(30,45,47,44);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,19)(18,23)(20,29)(21,32)(22,33)(24,39)(25,44)(26,40)(27,41)(28,37)(30,34)(31,38)(35,47)(36,48)(42,46)(43,45);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,20)(18,24)(19,29)(21,34)(22,36)(23,39)(25,37)(26,42)(27,43)(28,44)(30,32)(31,47)(33,48)(35,38)(40,46)(41,45);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,20)(18,25)(19,30)(21,35)(22,36)(23,39)(24,37)(26,43)(27,42)(28,45)(29,32)(31,48)(33,47)(34,38)(40,46)(41,44);(2,8)(6,9)(7,16)(14,15)(18,26)(23,40)(24,42)(25,43)(27,37)(28,41)(39,46)(44,45)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=h,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,f^b=fi,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, 7031320);