#7663170

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

6683 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=h,d^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^c=dhi,f^e=fhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7570366);