#8075517

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

4812 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gi,d^2=h,f^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, 7982713);