#7655112

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	#35165	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4934 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=hi,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7562308);