#7663155

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

6558 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)(5,6,10,8)(7,11,13,14)(9,12,17,16)(15,18,19,20)(21,22,28,32)(23,27,31,36)(24,25,35,29)(26,30,34,33);(1,2)(3,4)(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,20)(17,19)(21,23,28,31)(22,27,32,36)(24,33,35,30)(25,26,29,34);(7,13)(11,14)(15,19)(18,20)(21,24,28,35)(22,29,32,25)(23,33,31,30)(26,27,34,36);(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,20)(17,19)(21,25,28,29)(22,24,32,35)(23,26,31,34)(27,33,36,30);(5,8,10,6)(7,14,13,11)(9,12,17,16)(15,18,19,20)(21,26)(22,30)(23,29)(24,27)(25,31)(28,34)(32,33)(35,36);(5,9,10,17)(6,12,8,16)(7,15,13,19)(11,18,14,20)(21,27)(22,31)(23,32)(24,34)(25,33)(26,35)(28,36)(29,30)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=h,e^2=i,f^2=i,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, 7570351);