#8063113

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4751 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)(10,13)(14,16)(17,18,23,25)(19,26,20,22)(21,24,30,32)(27,31,28,29);(1,3)(2,4)(5,7)(6,8)(9,10,12,13)(11,14,15,16);(3,7)(4,8)(9,11,12,15)(10,14,13,16)(17,19,23,20)(18,22,25,26)(21,27,30,28)(24,29,32,31);(2,6)(4,8)(17,20,23,19)(18,22,25,26)(21,27,30,28)(24,31,32,29);(1,4)(2,3)(5,8)(6,7)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,32)(26,31);(1,2)(3,4)(5,6)(7,8)(17,22)(18,19)(20,25)(21,29)(23,26)(24,27)(28,32)(30,31)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=h,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, 7970309);