#8070407

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

7072 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)(7,10)(8,12)(9,13)(11,14)(15,18)(16,17)(19,20)(21,22)(23,30)(24,27)(25,28)(26,29)(31,35)(32,34)(33,36);(1,2)(3,4)(5,7)(6,10)(8,15)(9,14)(11,13)(12,18)(16,19)(17,20)(21,23,26,24)(22,27,29,30)(25,31,33,32)(28,34,36,35);(7,14)(10,11)(15,20)(18,19)(21,23,26,24)(22,27,29,30)(25,32,33,31)(28,35,36,34);(5,6)(7,10)(8,16)(9,13)(11,14)(12,17)(15,19)(18,20)(21,24,26,23)(22,27,29,30)(25,32,33,31)(28,34,36,35);(5,7)(6,11)(8,15)(9,14)(10,13)(12,19)(16,18)(17,20)(21,22)(23,27)(24,30)(25,28)(26,29)(31,34)(32,35)(33,36);(5,8)(6,12)(7,15)(9,17)(10,18)(11,19)(13,16)(14,20)(21,25)(22,28)(23,31)(24,32)(26,33)(27,34)(29,36)(30,35)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=h,d^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, 7977603);