#7748704

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

4745 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)(9,10,13,12)(11,15,16,14)(17,18,22,23)(19,26,24,20)(21,25,29,30)(27,32,31,28);(1,3,5,8)(2,6,4,7)(9,11)(10,14)(12,15)(13,16)(17,18)(19,20)(21,25)(22,23)(24,26)(27,28)(29,30)(31,32);(1,2,5,4)(3,6,8,7)(10,12)(14,15)(18,23)(19,24)(21,29)(28,32);(9,12,13,10)(11,15,16,14)(17,19,22,24)(18,20,23,26)(21,27,29,31)(25,28,30,32);(1,2,5,4)(3,6,8,7)(9,11)(10,14)(12,15)(13,16)(17,20)(18,24)(19,23)(21,28)(22,26)(25,31)(27,30)(29,32);(1,4,5,2)(3,7,8,6)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7655900);