#7656817

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	#34562	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6622 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)(5,6,10,13)(7,14,17,11)(8,9,12,18)(15,20,19,16)(21,22,26,27)(23,31,34,28)(24,29,25,30)(32,36,33,35);(1,2)(3,4)(22,27)(28,31)(29,30)(35,36);(5,7,10,17)(6,11,13,14)(8,15,12,19)(9,16,18,20)(21,22)(23,28)(24,29)(25,30)(26,27)(31,34)(32,35)(33,36);(21,23)(22,28)(24,32)(25,33)(26,34)(27,31)(29,35)(30,36);(5,8)(6,9)(7,15)(10,12)(11,16)(13,18)(14,20)(17,19)(21,24,26,25)(22,29,27,30)(23,32,34,33)(28,35,31,36);(5,9)(6,12)(7,16)(8,13)(10,18)(11,19)(14,15)(17,20)(21,25,26,24)(22,30,27,29)(23,33,34,32)(28,36,31,35)

Polycyclic presentation:

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

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, 7564013);