#7670807

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	#35463	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5428 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)(4,5)(6,8)(9,10)(11,15)(12,13)(14,16)(18,23)(21,29)(24,30)(26,32);(1,3,6,4)(2,7,8,5)(9,11)(10,14)(12,15)(13,16)(17,18)(19,24)(20,21)(22,23)(25,26)(27,30)(28,29)(31,32);(1,4,6,3)(2,7,8,5)(9,12,13,10)(11,15,16,14)(18,23)(20,28)(24,30)(25,31);(1,5,6,7)(2,4,8,3)(9,11)(10,15)(12,14)(13,16)(17,19,22,27)(18,24,23,30)(20,25,28,31)(21,26,29,32);(17,20)(18,21)(19,25)(22,28)(23,29)(24,26)(27,31)(30,32);(17,21)(18,20)(19,26)(22,29)(23,28)(24,25)(27,32)(30,31)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7578003);