#7712748

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	#37316	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:

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

Polycyclic presentation:

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

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, 7619944);