#7897659

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
prime radical	#41950	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4627 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)(17,18,23,22)(19,27,29,24)(20,30,31,25)(21,32,28,26);(1,2,4,5)(3,6,7,8)(17,19,23,29)(18,24,22,27)(20,28,31,21)(25,26,30,32);(9,10,13,12)(11,15,16,14)(17,20,23,31)(18,25,22,30)(19,21,29,28)(24,32,27,26);(9,11,13,16)(10,14,12,15)(17,20)(18,25)(19,28)(21,29)(22,30)(23,31)(24,26)(27,32);(9,12,13,10)(11,15,16,14)(17,21,23,28)(18,26,22,32)(19,20,29,31)(24,30,27,25);(1,3,4,7)(2,6,5,8)(17,22,23,18)(19,27,29,24)(20,30,31,25)(21,26,28,32)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=hi,e^2=h,f^2=g,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, 7804855);