#7658892

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

6844 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 7566088);