#8664671

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	#36362	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4867 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=i,d^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=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, 8571867);