#7757968

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6474 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:

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

Polycyclic presentation:

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

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, 7665164);