#7780152

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	#37658	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4833 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,d^2=gh,e^2=h,f^2=gi,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, 7687348);