#7633228

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4864 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=i,d^2=h,e^2=h,f^2=hi,b^a=bg,c^a=ch,c^b=ci,f^b=fi,e^d=ei

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7540424);