#8254697

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	#42895	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4539 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,d^2=h,e^2=hi,f^2=gi,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, 8161893);