#7979857

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	C₂×#547	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4659 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=i,e^2=g,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, 7887053);