#7659014

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

6722 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)(8,16)(11,17)(15,19)(18,20)(21,22,26,25)(23,29,30,27)(24,32,33,28)(31,34,36,35);(1,2,3,4)(5,6)(7,14)(8,17)(9,12)(10,13)(11,16)(15,18)(19,20)(23,30)(27,29)(31,36)(34,35);(5,7,10,12)(6,9,13,14)(8,15,16,19)(11,18,17,20)(21,23)(22,27)(24,31)(25,29)(26,30)(28,34)(32,35)(33,36);(5,8)(6,11)(7,15)(9,18)(10,16)(12,19)(13,17)(14,20)(21,24,26,33)(22,28,25,32)(23,31,30,36)(27,34,29,35);(5,6)(7,9)(8,11)(10,13)(12,14)(15,18)(16,17)(19,20)(21,25,26,22)(23,29,30,27)(24,32,33,28)(31,35,36,34);(5,9)(6,12)(7,13)(8,18)(10,14)(11,19)(15,17)(16,20)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=hi,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7566210);