#7919684

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

4918 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)(3,8)(10,14)(15,16)(17,18,23,27)(19,26,28,22)(20,24,31,32)(21,29,30,25);(1,2)(3,4)(5,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,29)(21,24)(23,28)(25,31)(26,27)(30,32);(1,3)(2,7)(4,6)(5,8)(17,20,23,31)(18,24,27,32)(19,25,28,29)(21,22,30,26);(1,2)(3,4)(5,6)(7,8)(9,11,13,12)(10,15,14,16)(17,21,23,30)(18,25,27,29)(19,24,28,32)(20,22,31,26);(1,2)(3,7)(4,8)(5,6)(9,11,13,12)(10,15,14,16)(17,22)(18,19)(20,30)(21,31)(23,26)(24,25)(27,28)(29,32);(1,4,5,7)(2,3,6,8)(9,12,13,11)(10,16,14,15)(17,19)(18,26)(20,25)(21,24)(22,27)(23,28)(29,31)(30,32)

Polycyclic presentation:

2^9|a^2=hi,c^2=hi,d^2=gh,e^2=gi,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, 7826880);