#7631190

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

4890 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=i,e^2=i,f^2=gh,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, 7538386);