#7987511

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

6875 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=hi,d^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,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, 7894707);