#7968275

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

5082 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,e^2=gh,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, 7875471);