#7785149

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	#37672	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5147 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)(20,25)(21,30)(22,24)(23,28)(26,31)(29,32);(1,2)(3,7)(4,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,28)(18,23,24,27)(20,21,29,31)(25,26,32,30);(2,5)(6,7)(18,24)(19,28)(20,29)(26,30);(3,8)(6,7)(9,11,12,15)(10,14,13,16)(17,20,22,29)(18,25,24,32)(19,21,28,31)(23,26,27,30);(1,3)(2,6)(4,8)(5,7)(17,21)(18,26)(19,20)(22,31)(23,25)(24,30)(27,32)(28,29);(1,2)(3,6)(4,5)(7,8)(17,18)(19,23)(20,25)(21,26)(22,24)(27,28)(29,32)(30,31)

Polycyclic presentation:

2^9|b^2=h,d^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, 7692345);