#7659679

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

6732 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)(21,22,26,29)(23,30,33,27)(24,25,28,34)(31,36,35,32);(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18);(6,10)(11,17)(12,18)(19,20)(21,23,26,33)(22,27,29,30)(24,31,28,35)(25,32,34,36);(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)(21,23)(22,27)(24,31)(25,32)(26,33)(28,35)(29,30)(34,36);(5,7,9,14)(6,11,10,17)(8,15,16,13)(12,20,18,19)(21,24)(22,25)(23,31)(26,28)(27,32)(29,34)(30,36)(33,35);(5,8,9,16)(6,12,10,18)(7,13,14,15)(11,19,17,20)(21,25)(22,28)(23,32)(24,29)(26,34)(27,35)(30,31)(33,36)

Polycyclic presentation:

2^9|a^2=h,c^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, 7566875);