#7855580

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	#59417	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4683 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7762776);