#7733827

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

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

5336 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7641023);