#7717610

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	#35091	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6536 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:

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

Polycyclic presentation:

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

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, 7624806);