#8057945

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, prime radical, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4113 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,d^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72268205, #72271515
Fitting subgroup of	#72268205
maximal subgroup of	#72268205, #72271508
maximal normal subgroup of	#72268205, #72271508
maximal quotient of	#72268208

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7965141);