#7917811

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

5134 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=ghi,e^2=i,f^2=ghi,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, 7825007);