#7685526

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	#59074	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4844 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=g,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, 7592722);