#7716001

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	#35003	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4710 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)(17,18,23,24)(19,22,21,30)(20,25,27,31)(26,29,28,32);(9,10,12,13)(11,14,15,16)(17,19)(18,22)(20,26)(21,23)(24,30)(25,29)(27,28)(31,32);(2,3)(6,7)(18,24)(19,21)(20,27)(29,32);(1,3,5,2)(4,7,8,6)(9,11,12,15)(10,14,13,16)(17,20,23,27)(18,25,24,31)(19,26,21,28)(22,29,30,32);(1,4,5,8)(2,6,3,7)(17,21,23,19)(18,22,24,30)(20,28,27,26)(25,29,31,32);(17,22)(18,19)(20,29)(21,24)(23,30)(25,26)(27,32)(28,31)

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=gh,e^2=h,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, 7623197);