#8325513

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	#39126	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4502 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:

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

Polycyclic presentation:

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

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, 8232709);