#7643016

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	#39165	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4726 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:

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

Polycyclic presentation:

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

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, 7550212);