#7850381

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	#2846	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5225 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,e^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7757577);