#7631152

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
prime radical	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4934 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)(10,15)(11,16)(20,28)(23,30)(26,32)(29,31);(1,2,5,4)(3,6,8,7)(9,10,14,15)(11,12,16,13)(17,18,21,24)(19,22,27,25)(20,23,28,30)(26,29,32,31);(1,3,5,8)(2,6,4,7)(9,11)(10,12)(13,15)(14,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32);(9,12,14,13)(10,16,15,11)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31);(9,12,14,13)(10,16,15,11)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32);(1,4,5,2)(3,7,8,6)(9,13,14,12)(10,11,15,16)(17,20)(18,23)(19,26)(21,28)(22,29)(24,30)(25,31)(27,32)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538348);