#7709449

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	#34986	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4870 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)(3,7)(4,6)(5,8)(10,13)(14,16)(18,23)(21,31)(24,32)(25,28);(1,3,6,8)(2,7,4,5)(9,10)(11,14)(12,13)(15,16)(17,18,22,23)(19,24,29,32)(20,25,30,28)(21,26,31,27);(2,4)(3,8)(18,23)(20,30)(24,32)(26,27);(1,3,6,8)(2,7,4,5)(9,11,12,15)(10,14,13,16)(17,19,22,29)(18,24,23,32)(20,27,30,26)(21,28,31,25);(1,4,6,2)(3,5,8,7)(17,20)(18,25)(19,27)(21,24)(22,30)(23,28)(26,29)(31,32);(1,5)(2,3)(4,8)(6,7)(17,21)(18,26)(19,28)(20,24)(22,31)(23,27)(25,29)(30,32)

Polycyclic presentation:

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

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, 7616645);