#7969364

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

4836 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,e^2=g,f^2=g,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7876560);