#7792882

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	#44270	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4646 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:

(9,10,13,16)(11,14,12,15)(17,18)(19,23)(20,26)(21,25)(22,24)(27,30)(28,32)(29,31);(1,2)(3,7)(4,5)(6,8)(9,10)(11,14)(12,15)(13,16)(17,19,22,26)(18,23,24,20)(21,27,29,32)(25,30,31,28);(2,5)(6,7)(9,11,13,12)(10,14,16,15)(18,24)(19,26)(21,29)(28,30);(3,8)(6,7)(9,11,13,12)(10,14,16,15)(17,18)(19,23)(20,26)(21,25)(22,24)(27,30)(28,32)(29,31);(1,3,4,8)(2,6,5,7)(9,11,13,12)(10,14,16,15)(17,20,22,23)(18,19,24,26)(21,28,29,30)(25,27,31,32);(1,2)(3,6)(4,5)(7,8)(9,12,13,11)(10,15,16,14)(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=h,c^2=g,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7700078);