#8390708

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	#38505	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4761 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8297904);