#7650685

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×#535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6652 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)(5,6,10,9)(7,11,13,15)(8,12,16,17)(14,18,19,20)(21,22)(23,27)(24,32)(25,28)(26,30)(29,33)(31,36)(34,35);(1,2)(3,4)(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,19)(17,20)(21,23,26,25)(22,27,30,28)(24,31,33,34)(29,35,32,36);(7,13)(11,15)(14,19)(18,20)(21,23,26,25)(22,28,30,27)(24,31,33,34)(29,36,32,35);(5,8)(6,12)(7,14)(9,17)(10,16)(11,18)(13,19)(15,20)(21,24)(22,29)(23,31)(25,34)(26,33)(27,35)(28,36)(30,32);(5,9,10,6)(7,15,13,11)(8,12,16,17)(14,18,19,20)(21,25,26,23)(22,28,30,27)(24,31,33,34)(29,35,32,36);(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,19)(17,20)(21,22)(23,27)(24,29)(25,28)(26,30)(31,35)(32,33)(34,36)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7557881);