#7806595

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	#34976	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6586 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)(5,6,10,9)(7,11,13,17)(8,12,15,16)(14,18,19,20)(21,22,26,27)(23,31,32,28)(24,29,25,30)(33,36,34,35);(1,2)(3,4)(6,9)(7,13)(8,15)(18,20)(22,27)(23,32)(24,25)(35,36);(5,7)(6,11)(8,14)(9,17)(10,13)(12,18)(15,19)(16,20)(21,23,26,32)(22,28,27,31)(24,34,25,33)(29,36,30,35);(5,8)(6,12)(7,14)(9,16)(10,15)(11,18)(13,19)(17,20)(21,24,26,25)(22,29,27,30)(23,33,32,34)(28,35,31,36);(5,8)(6,12)(7,14)(9,16)(10,15)(11,18)(13,19)(17,20)(21,22)(23,28)(24,29)(25,30)(26,27)(31,32)(33,35)(34,36);(5,9,10,6)(7,11,13,17)(8,16,15,12)(14,18,19,20)(21,25,26,24)(22,30,27,29)(23,34,32,33)(28,36,31,35)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,e^b=ei,f^b=fi,d^c=dh,f^c=fi

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, 7713791);