#8081022

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	#43023	C₂
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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7988218);