#7969175

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

4869 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)(9,10,13,12)(11,15,16,14)(17,18)(19,25)(20,22)(21,24)(23,26)(27,32)(28,30)(29,31);(1,3)(2,4)(5,6)(7,8)(9,11,13,16)(10,14,12,15)(17,19)(18,23)(20,25)(21,29)(22,26)(24,32)(27,30)(28,31);(1,4)(2,6)(3,8)(5,7)(9,10,13,12)(11,14,16,15)(18,20)(19,26)(24,28)(27,29);(1,3)(2,7)(4,8)(5,6)(17,19)(18,23)(20,25)(21,27)(22,26)(24,31)(28,32)(29,30);(1,3)(2,4)(5,6)(7,8)(17,20,22,18)(19,25,26,23)(21,24,30,28)(27,31,29,32);(1,4)(2,6)(3,8)(5,7)(9,12,13,10)(11,15,16,14)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,32)(26,29)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gh,e^2=h,f^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, 7876371);