#7670929

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×#121	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4881 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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=g,d^2=h,e^2=h,f^2=g,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, 7578125);