#7124129

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	#61270	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)(18,24)(19,28)(21,22)(26,38)(27,35)(29,43)(30,31)(32,42)(33,36)(37,40)(44,47)(45,46);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,23,27)(19,25,31,41)(20,32,34,38)(21,24,22,35)(26,36,42,33)(28,40,30,37)(29,44,46,47)(39,43,48,45);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19)(18,25)(20,29)(21,30)(22,28)(23,31)(24,37)(26,39)(27,41)(32,44)(33,45)(34,46)(35,40)(36,43)(38,47)(42,48);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,20)(18,26)(19,29)(21,33)(22,36)(23,34)(24,38)(25,39)(27,42)(28,43)(30,45)(31,46)(32,35)(37,47)(40,44)(41,48);(2,8)(6,9)(7,16)(14,15)(17,21,23,22)(18,24,27,35)(19,30,31,28)(20,33,34,36)(25,37,41,40)(26,38,42,32)(29,45,46,43)(39,47,48,44)

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=h,e^2=i,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, 7031325);