#8214528

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	#37649	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4745 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=g,e^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8121724);