#7632193

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

5099 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7539389);