#7728983

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

4776 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=h,d^2=hi,e^2=hi,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=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, 7636179);