#7690021

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4601 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,d^2=h,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, 7597217);