#7690848

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

4821 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)(10,13)(14,15)(17,18)(19,28)(20,22)(21,26)(23,27)(24,30)(25,31)(29,32);(1,3,5,8)(2,6,4,7)(17,19,23,30)(18,24,27,28)(20,29,31,21)(22,26,25,32);(9,10,12,13)(11,15,16,14)(17,20,23,31)(18,25,27,22)(19,21,30,29)(24,26,28,32);(1,2,5,4)(3,6,8,7)(9,11,12,16)(10,14,13,15)(17,20)(18,22)(19,29)(21,30)(23,31)(24,26)(25,27)(28,32);(1,4,5,2)(3,7,8,6)(17,21,23,29)(18,26,27,32)(19,20,30,31)(22,24,25,28);(17,22)(18,20)(19,26)(21,28)(23,25)(24,29)(27,31)(30,32)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,d^2=gh,e^2=g,b^a=bg,c^a=ch,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7598044);