#7720378

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	#35236	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4780 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7627574);