#7715654

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

6498 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei

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, 7622850);