#7943372

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

4892 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,d^2=gi,e^2=gi,f^2=ghi,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, 7850568);