#8434322

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	#36098	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5307 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8341518);