#8063143

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	#52031	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4957 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)(3,4)(5,6)(7,8)(10,14)(15,16)(17,18,22,24)(19,25,29,23)(20,26,31,21)(27,28,30,32);(1,3)(2,4)(5,7)(6,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,27)(21,28)(22,29)(24,25)(26,32)(30,31);(3,7)(4,8)(9,11,13,12)(10,15,14,16)(18,24)(21,26)(23,25)(28,32);(2,6)(4,8)(9,11,13,12)(10,15,14,16)(17,19)(18,25)(20,30)(21,28)(22,29)(23,24)(26,32)(27,31);(1,4)(2,3)(5,8)(6,7)(17,20)(18,26)(19,27)(21,24)(22,31)(23,32)(25,28)(29,30);(1,2)(3,4)(5,6)(7,8)(9,12,13,11)(10,16,14,15)(17,21,22,26)(18,20,24,31)(19,28,29,32)(23,27,25,30)

Polycyclic presentation:

2^9|a^2=h,c^2=gh,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,c^b=ci,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, 7970339);