#7806596

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

6744 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)(5,6,10,13)(7,11,14,19)(8,12,17,9)(15,16,20,18)(22,27)(24,32)(28,30)(31,33);(1,2,3,4)(7,14)(8,17)(9,12)(11,19)(22,27)(23,25)(24,32)(35,36);(5,7)(6,11)(8,15)(9,18)(10,14)(12,16)(13,19)(17,20)(21,22,26,27)(23,30,25,28)(24,29,32,34)(31,36,33,35);(5,8)(6,12)(7,15)(9,13)(10,17)(11,16)(14,20)(18,19)(21,23,26,25)(22,28,27,30)(24,31,32,33)(29,35,34,36);(5,8)(6,12)(7,15)(9,13)(10,17)(11,16)(14,20)(18,19)(21,24)(22,29)(23,31)(25,33)(26,32)(27,34)(28,35)(30,36);(5,9,10,12)(6,8,13,17)(7,16,14,18)(11,20,19,15)(21,25,26,23)(22,30,27,28)(24,33,32,31)(29,36,34,35)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7713792);