#7987198

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
prime radical	#2537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6337 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7894394);