#8075403

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

4888 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=g,e^2=h,b^a=bg,c^a=ch,e^a=ei,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, 7982599);