#7859928

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4865 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gi,d^2=g,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, 7767124);