#7785173

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	#43886	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5073 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:

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

Polycyclic presentation:

2^9|b^2=h,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, 7692369);