#7983256

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	#45478	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4789 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7890452);