#7772098

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	#41982	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4737 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)(20,29)(21,26)(22,24)(25,28)(27,32)(30,31);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19,22,27)(18,23,24,32)(20,21,28,30)(25,31,29,26);(2,5)(6,8)(18,24)(19,27)(21,30)(25,29);(3,7)(6,8)(9,11,12,15)(10,13,14,16)(17,18)(19,23)(20,29)(21,26)(22,24)(25,28)(27,32)(30,31);(1,3,4,7)(2,6,5,8)(17,20,22,28)(18,25,24,29)(19,21,27,30)(23,31,32,26);(1,2)(3,6)(4,5)(7,8)(17,21)(18,26)(19,28)(20,27)(22,30)(23,25)(24,31)(29,32)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=g,e^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,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, 7679294);