#7952525

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

6689 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)(8,13)(9,12)(11,16)(14,18)(15,20)(17,19)(21,22)(23,29)(24,28)(25,27)(26,31)(30,35)(32,34)(33,36);(1,2,3,4)(5,6,10,7)(8,11,16,13)(9,12,18,14)(15,19,17,20)(21,23)(22,26)(24,30)(25,31)(27,29)(28,33)(32,36)(34,35);(6,7)(11,13)(12,14)(19,20)(22,27)(26,29)(28,34)(33,35);(5,7,10,6)(8,13,16,11)(9,14,18,12)(15,20,17,19)(21,22)(23,26)(24,28)(25,27)(29,31)(30,33)(32,34)(35,36);(5,8)(6,11)(7,13)(9,17)(10,16)(12,20)(14,19)(15,18)(21,23)(22,26)(24,30)(25,31)(27,29)(28,33)(32,36)(34,35);(5,9,10,18)(6,12,7,14)(8,15,16,17)(11,19,13,20)(21,24,25,32)(22,28,27,34)(23,30,31,36)(26,33,29,35)

Polycyclic presentation:

2^9|b^2=ghi,d^2=i,f^2=h,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, 7859721);