#7787289

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	#43118	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4921 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)(11,15)(12,13)(14,16)(17,18)(19,28)(20,25)(21,26)(22,24)(23,27)(29,32)(30,31);(1,2)(3,7)(4,8)(5,6)(17,19,23,22)(18,24,27,28)(20,29,26,31)(21,32,25,30);(10,13)(14,15)(19,22)(24,28)(29,31)(30,32);(9,10)(11,14)(12,13)(15,16)(17,20)(18,25)(19,29)(21,27)(22,31)(23,26)(24,30)(28,32);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,16)(13,15)(17,21,23,25)(18,26,27,20)(19,30,22,32)(24,29,28,31);(1,4,5,3)(2,8,6,7)(17,22,23,19)(18,24,27,28)(20,31,26,29)(21,32,25,30)

Polycyclic presentation:

2^9|b^2=i,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=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, 7694485);