#8091722

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

4905 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,e^2=ghi,f^2=gi,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, 7998918);