#7124109

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	#3493	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1565 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)(17,18,24,29)(19,30,25,35)(20,32,36,33)(21,26,40,45)(22,28,23,27)(31,47,42,43)(34,37,38,48)(39,46,41,44);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,19)(18,20)(21,37)(22,33)(23,32)(24,25)(26,43)(27,30)(28,35)(29,36)(31,39)(34,44)(38,46)(40,48)(41,42)(45,47);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,20)(18,25)(19,29)(21,38)(22,35)(23,30)(24,36)(26,31)(27,33)(28,32)(34,40)(37,44)(39,47)(41,43)(42,45)(46,48);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,21,24,40)(18,26,29,45)(19,31,25,42)(20,34,36,38)(22,39,23,41)(27,44,28,46)(30,47,35,43)(32,37,33,48);(2,8)(6,9)(7,16)(14,15)(17,22,24,23)(18,27,29,28)(19,32,25,33)(20,35,36,30)(21,39,40,41)(26,44,45,46)(31,37,42,48)(34,43,38,47)

Polycyclic presentation:

2^9|a^2=i,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, 7031305);