#7643915

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

6740 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)(22,26)(23,29)(28,33)(30,35);(1,2)(3,4)(5,6)(7,13)(8,12)(9,10)(11,15)(14,19)(16,17)(18,20)(22,26)(27,32)(28,33)(34,36);(6,10)(11,13)(12,17)(18,19)(21,22,25,26)(23,27,29,32)(24,28,31,33)(30,34,35,36);(5,7,9,15)(6,11,10,13)(8,14,16,20)(12,18,17,19)(21,23,25,29)(22,27,26,32)(24,30,31,35)(28,34,33,36);(5,6)(7,11)(8,12)(9,10)(13,15)(14,18)(16,17)(19,20)(21,24,25,31)(22,28,26,33)(23,30,29,35)(27,34,32,36);(5,8)(6,12)(7,14)(9,16)(10,17)(11,18)(13,19)(15,20)

Polycyclic presentation:

2^9|c^2=h,d^2=i,e^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, 7551111);