#8069509

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

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

6495 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 36 points:

(1,2,3,4)(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)(22,28)(23,24)(30,35)(31,32);(1,2)(3,4)(21,22)(23,25)(24,29)(26,30)(27,28)(31,33)(32,36)(34,35);(6,10)(11,17)(12,18)(19,20)(21,23,27,24)(22,29,28,25)(26,32,34,31)(30,33,35,36);(5,7,9,14)(6,11,10,17)(8,15,16,13)(12,20,18,19)(21,24,27,23)(22,29,28,25)(26,31,34,32)(30,33,35,36);(5,8,9,16)(6,12,10,18)(7,13,14,15)(11,19,17,20)(21,25,27,29)(22,23,28,24)(26,33,34,36)(30,31,35,32);(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)(21,26,27,34)(22,30,28,35)(23,31,24,32)(25,33,29,36)

Polycyclic presentation:

2^9|a^2=gi,c^2=i,d^2=h,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=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, 7976705);