#7642032

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
prime radical	C₂²×#538	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7326 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 36 points:

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

Polycyclic presentation:

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

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, 7549228);