#8071527

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

6970 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)(7,10)(8,12)(9,13)(11,14)(15,18)(16,17)(19,20)(21,22)(23,28)(24,26)(25,30)(27,31)(29,34)(32,36)(33,35);(1,2,3,4)(5,7)(6,10)(8,15)(9,14)(11,13)(12,18)(16,19)(17,20)(21,23)(22,28)(24,25)(26,30)(27,33)(29,36)(31,35)(32,34);(7,14)(10,11)(15,20)(18,19)(21,24,27,34)(22,29,31,26)(23,32,33,25)(28,30,35,36);(5,6)(7,10)(8,16)(9,13)(11,14)(12,17)(15,19)(18,20)(21,25)(22,30)(23,24)(26,28)(27,32)(29,35)(31,36)(33,34);(5,7)(6,11)(8,15)(9,14)(10,13)(12,19)(16,18)(17,20)(21,24)(22,29)(23,25)(26,31)(27,34)(28,36)(30,35)(32,33);(5,8)(6,12)(7,15)(9,17)(10,18)(11,19)(13,16)(14,20)(21,26,27,29)(22,24,31,34)(23,30,33,36)(25,35,32,28)

Polycyclic presentation:

2^9|b^2=g,c^2=h,f^2=h,b^a=bg,c^a=ch,e^a=ei,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, 7978723);