#7985154

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	#59023	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5463 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

(1,2,5,6)(3,7,8,4)(9,10)(11,12)(13,15)(14,16)(17,18,22,20)(19,25,27,23)(21,24,29,28)(26,31,32,30);(2,6)(3,8)(9,11)(10,12)(13,16)(14,15)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32);(1,3)(2,7)(4,6)(5,8)(9,12,14,13)(10,11,16,15)(18,20)(23,25)(24,28)(30,31);(1,4)(2,8)(3,6)(5,7)(9,11)(10,13)(12,16)(14,15)(17,20,22,18)(19,25,27,23)(21,28,29,24)(26,31,32,30);(1,4)(2,3)(5,7)(6,8)(9,11)(10,12)(13,16)(14,15)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32);(1,3)(2,7)(4,6)(5,8)(9,13,14,12)(10,15,16,11)(17,21,22,29)(18,24,20,28)(19,26,27,32)(23,30,25,31)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72267714, #72267729, #72267730

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7892350);