#7927106

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

5227 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)(9,10,13,12)(11,15,16,14)(17,18,23,24)(19,21,27,31)(20,29,28,25)(22,26,32,30);(1,3)(2,6)(4,7)(5,8)(9,11,13,16)(10,14,12,15)(17,19,23,27)(18,21,24,31)(20,22,28,32)(25,30,29,26);(2,4)(6,7)(18,24)(19,27)(20,28)(26,30);(1,4,5,2)(3,7,8,6)(17,20,23,28)(18,25,24,29)(19,22,27,32)(21,30,31,26);(1,3)(2,6)(4,7)(5,8)(9,12,13,10)(11,15,16,14)(17,21,23,31)(18,19,24,27)(20,30,28,26)(22,29,32,25);(17,22)(18,26)(19,28)(20,27)(21,25)(23,32)(24,30)(29,31)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=h,e^2=gh,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, 7834302);