#7784299

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

4988 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:

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

Polycyclic presentation:

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

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, 7691495);