#7901253

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
prime radical	C₂×#547	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4617 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7808449);