#7124108

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

1580 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)(18,24)(19,28)(21,22)(26,40)(27,37)(30,45)(31,38)(32,44)(33,47)(34,35)(39,42)(46,48);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,29)(20,32)(21,37)(22,24)(23,27)(25,31)(26,35)(28,39)(30,46)(33,48)(34,44)(36,40)(38,42)(41,45)(43,47);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19)(18,25)(20,33)(21,28)(22,38)(23,31)(24,39)(26,43)(27,29)(30,36)(32,46)(34,47)(35,45)(37,42)(40,48)(41,44);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,20,23,36)(18,26,27,44)(19,30,31,33)(21,34,22,35)(24,40,37,32)(25,41,29,43)(28,45,38,47)(39,46,42,48);(2,8)(6,9)(7,16)(14,15)(17,21,23,22)(18,24,27,37)(19,28,31,38)(20,34,36,35)(25,39,29,42)(26,40,44,32)(30,45,33,47)(41,46,43,48)

Polycyclic presentation:

2^9|c^2=hi,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,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, 7031304);