#7892203

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

6803 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)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22)(23,30)(24,32)(25,27)(26,29)(28,34)(31,35)(33,36);(1,2,3,4)(21,23,26,25)(22,27,29,30)(24,31,34,33)(28,35,32,36);(5,6,9,12)(7,10,15,13)(8,11,16,18)(14,17,20,19)(21,22)(23,30)(24,28)(25,27)(26,29)(31,36)(32,34)(33,35);(5,6)(7,10)(8,11)(9,12)(13,15)(14,17)(16,18)(19,20)(21,24,26,34)(22,28,29,32)(23,31,25,33)(27,35,30,36);(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(21,22)(23,27)(24,28)(25,30)(26,29)(31,35)(32,34)(33,36);(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(21,25,26,23)(22,27,29,30)(24,33,34,31)(28,35,32,36)

Polycyclic presentation:

2^9|b^2=g,c^2=h,d^2=i,f^2=hi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=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, 7799399);