#7785146

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	#36704	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4958 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^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, 7692342);