#7738058

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	#37836	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4806 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=g,e^2=g,f^2=i,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, 7645254);