#7715437

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	#39335	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4614 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7622633);