#7690819

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	#36468	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4677 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gh,d^2=h,e^2=g,f^2=g,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, 7598015);