#7661811

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×#3521	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6668 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^c=dhi,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, 7569007);