#7919560

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

6628 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=hi,c^2=hi,e^2=h,f^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, 7826756);