#7756812

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

4989 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,e^2=ghi,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,e^b=ei,d^c=dh,f^c=fi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7664008);