#8551044

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#57396	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4041 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=h,d^2=h,e^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8458240);