#7642156

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	C₂×#3455	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6550 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)(12,17)(14,20)(18,19)(23,29)(24,32)(26,30)(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)(23,30,29,26)(24,27,32,34)(31,36,35,33);(6,10)(11,13)(12,17)(18,19)(21,23,25,29)(22,26,28,30)(24,31,32,35)(27,33,34,36);(5,6)(7,11)(8,12)(9,10)(13,15)(14,18)(16,17)(19,20)(21,24)(22,27)(23,31)(25,32)(26,33)(28,34)(29,35)(30,36);(5,7,9,15)(6,11,10,13)(8,14,16,20)(12,18,17,19);(5,8)(6,12)(7,14)(9,16)(10,17)(11,18)(13,19)(15,20)

Polycyclic presentation:

2^9|b^2=h,c^2=h,e^2=hi,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, 7549352);