#8070358

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	#37654	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4728 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)(9,10,13,16)(11,14,12,15)(17,18,23,25)(19,27,28,24)(20,21,30,31)(22,26,32,29);(1,3)(2,4)(5,8)(6,7)(9,10)(11,14)(12,15)(13,16)(17,19,23,28)(18,24,25,27)(20,22,30,32)(21,29,31,26);(1,2,5,6)(3,7,8,4)(9,11,13,12)(10,14,16,15)(18,25)(19,28)(21,31)(22,32);(1,2,5,6)(3,4,8,7)(9,12,13,11)(10,15,16,14)(17,20)(18,21)(19,22)(23,30)(24,29)(25,31)(26,27)(28,32);(1,2)(3,4)(5,6)(7,8)(17,21)(18,20)(19,29)(22,24)(23,31)(25,30)(26,28)(27,32);(1,4)(2,3)(5,7)(6,8)(17,22)(18,26)(19,30)(20,28)(21,24)(23,32)(25,29)(27,31)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=gh,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, 7977554);