#7684836

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	#44651	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4798 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)(3,4)(5,8)(6,7)(10,14)(15,16)(17,18)(19,29)(20,25)(21,26)(22,27)(23,28)(24,32)(30,31);(9,10,13,14)(11,15,12,16)(17,19)(18,22)(20,21)(23,27)(24,26)(25,31)(28,29)(30,32);(2,7)(3,8)(17,20,23,32)(18,24,28,25)(19,30,27,21)(22,31,29,26);(1,3,6,8)(2,4,7,5)(9,11,13,12)(10,15,14,16)(17,20)(18,25)(19,21)(22,31)(23,32)(24,28)(26,29)(27,30);(1,4,6,5)(2,3,7,8)(9,12,13,11)(10,16,14,15)(17,21)(18,26)(19,20)(22,24)(23,30)(25,29)(27,32)(28,31);(1,5,6,4)(2,8,7,3)(17,22,23,29)(18,27,28,19)(20,31,32,26)(21,25,30,24)

Polycyclic presentation:

2^9|b^2=gi,c^2=i,d^2=gh,e^2=gh,f^2=h,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, 7592032);