#7644359

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

6860 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)(5,6)(7,13)(8,16)(9,10)(11,14)(12,17)(15,18)(19,20)(25,33)(28,32)(30,36)(34,35);(1,2)(3,4)(7,14)(11,13)(15,20)(18,19)(21,22,26,24)(23,29,31,27)(25,28,33,32)(30,35,36,34);(6,10)(11,13)(12,16)(18,19)(22,24)(27,29)(28,32)(34,35);(5,7)(6,11)(8,15)(9,14)(10,13)(12,18)(16,19)(17,20)(21,23,26,31)(22,27,24,29)(25,30,33,36)(28,34,32,35);(5,6)(7,11)(8,12)(9,10)(13,14)(15,18)(16,17)(19,20)(21,24,26,22)(23,29,31,27)(25,32,33,28)(30,35,36,34);(5,8,9,17)(6,12,10,16)(7,15,14,20)(11,18,13,19)(21,25)(22,28)(23,30)(24,32)(26,33)(27,34)(29,35)(31,36)

Polycyclic presentation:

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

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, 7551555);