#7687056

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	#44151	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4702 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=i,e^2=hi,f^2=gi,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, 7594252);