#8077405

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

5297 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

Polycyclic presentation:

2^9|a^2=gh,c^2=gh,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,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, 7984601);