#7876767

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

6720 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)(5,6,11,12)(7,10,15,20)(8,13,17,19)(9,18,16,14)(21,22,26,30)(23,29,34,27)(24,28,35,25)(31,36,33,32);(1,2,3,4)(5,7,11,15)(6,10,12,20)(8,16,17,9)(13,14,19,18)(21,23)(22,27)(24,33)(25,32)(26,34)(28,36)(29,30)(31,35);(6,12)(7,15)(8,17)(14,18)(21,24)(22,28)(23,31)(25,30)(26,35)(27,32)(29,36)(33,34);(5,8)(6,13)(7,16)(9,15)(10,14)(11,17)(12,19)(18,20)(21,25,26,28)(22,24,30,35)(23,32,34,36)(27,33,29,31);(5,9,11,16)(6,14,12,18)(7,8,15,17)(10,19,20,13)(21,24)(22,28)(23,33)(25,30)(26,35)(27,36)(29,32)(31,34);(5,10)(6,15)(7,12)(8,14)(9,13)(11,20)(16,19)(17,18)(21,23)(22,29)(24,31)(25,32)(26,34)(27,30)(28,36)(33,35)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,d^2=hi,e^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,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, 7783963);