#7654865

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	#34976	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4754 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=hi,d^2=h,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7562061);