#7658217

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

7008 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)(7,12)(8,18)(9,14)(10,15)(11,16)(13,19)(17,20)(21,22,28,27)(23,32,30,25)(24,29,36,35)(26,31,34,33);(1,2)(3,4)(5,7,11,9)(6,12,16,14)(8,17,19,10)(13,15,18,20)(21,23)(22,25)(24,31)(26,29)(27,32)(28,30)(33,36)(34,35);(5,8,11,19)(6,13,16,18)(7,17,9,10)(12,15,14,20)(21,24)(22,29)(23,33)(25,34)(26,32)(27,35)(28,36)(30,31);(5,9,11,7)(6,14,16,12)(8,10,19,17)(13,20,18,15)(21,25,28,32)(22,30,27,23)(24,34,36,26)(29,31,35,33);(5,10)(6,15)(7,8)(9,19)(11,17)(12,18)(13,14)(16,20)(21,26,28,34)(22,31,27,33)(23,29,30,35)(24,32,36,25);(5,6)(7,12)(8,13)(9,14)(10,20)(11,16)(15,17)(18,19)(21,27,28,22)(23,32,30,25)(24,35,36,29)(26,31,34,33)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=hi,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,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, 7565413);