#7658016

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	#34624	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6452 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)(5,6,9,12)(7,10,14,18)(8,11,16,20)(13,17,15,19)(22,27)(23,31)(29,30)(32,33);(1,2,3,4)(22,27)(28,34)(29,30)(35,36);(5,6)(7,10)(8,11)(9,12)(13,17)(14,18)(15,19)(16,20)(21,22)(23,28)(24,29)(25,30)(26,27)(31,34)(32,35)(33,36);(21,23,26,31)(22,28,27,34)(24,32,25,33)(29,35,30,36);(5,7,9,14)(6,10,12,18)(8,15,16,13)(11,19,20,17)(21,24,26,25)(22,29,27,30)(23,32,31,33)(28,35,34,36);(5,8,9,16)(6,11,12,20)(7,13,14,15)(10,17,18,19)(21,25,26,24)(22,30,27,29)(23,33,31,32)(28,36,34,35)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,d^2=i,e^2=h,f^2=h,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, 7565212);