#7877806

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	#49578	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4704 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=gh,d^2=ghi,f^2=gh,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, 7785002);