#7920209

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

4682 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=h,e^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=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, 7827405);