#7643532

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×#3512	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6590 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)(5,6,10,11)(7,13,15,8)(9,17,18,12)(14,16,20,19)(24,31)(28,33)(29,35)(34,36);(1,2,3,4)(21,22)(23,27)(24,28)(25,26)(29,34)(30,32)(31,33)(35,36);(6,11)(8,13)(12,17)(16,19)(22,26)(27,32)(28,33)(34,36);(5,7)(6,8)(9,14)(10,15)(11,13)(12,16)(17,19)(18,20)(21,23,25,30)(22,27,26,32)(24,29,31,35)(28,34,33,36);(5,8)(6,7)(9,16)(10,13)(11,15)(12,14)(17,20)(18,19)(21,22)(23,27)(24,28)(25,26)(29,34)(30,32)(31,33)(35,36);(5,9,10,18)(6,12,11,17)(7,14,15,20)(8,16,13,19)(21,24,25,31)(22,28,26,33)(23,29,30,35)(27,34,32,36)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7550728);