#8075550

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

6472 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)(7,8)(9,13)(10,12)(11,14)(15,16)(17,19)(18,20)(21,22)(23,30)(24,27)(25,28)(26,29)(31,35)(32,34)(33,36);(1,2)(3,4)(5,7)(6,11)(8,12)(9,15)(10,14)(13,18)(16,19)(17,20);(6,12)(7,14)(9,17)(16,18)(21,23,26,24)(22,27,29,30)(25,32,33,31)(28,35,36,34);(5,8,10,11)(6,7,12,14)(9,18,17,16)(13,20,19,15)(21,24,26,23)(22,27,29,30)(25,32,33,31)(28,34,36,35);(5,7)(6,11)(8,12)(9,15)(10,14)(13,18)(16,19)(17,20)(21,22)(23,27)(24,30)(25,28)(26,29)(31,34)(32,35)(33,36);(5,9,10,17)(6,13,12,19)(7,15,14,20)(8,16,11,18)(21,25,26,33)(22,28,29,36)(23,31,24,32)(27,34,30,35)

Polycyclic presentation:

2^9|c^2=hi,d^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, 7982746);