#391609

order	512
nilpotency class	3
isoclinism	isoclinic class of #232181
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

696 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(9,10,15,20,16,21,14,32)(11,22,25,17,27,34,26,35)(12,28,30,18,13,19,31,37)(23,38,24,39,40,33,29,36);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,29)(13,24)(14,26)(15,25)(16,27)(18,36)(19,38)(20,34)(21,35)(22,32)(23,30)(28,33)(31,40)(37,39);(2,5)(6,8)(9,12,14,30,16,13,15,31)(10,18,20,28,21,37,32,19)(11,23,25,29,27,40,26,24)(17,33,22,36,35,38,34,39);(1,3,4,7)(2,6,5,8)(9,13)(10,19)(11,24)(12,16)(14,30)(15,31)(17,33)(18,32)(20,37)(21,28)(22,39)(23,26)(25,40)(27,29)(34,36)(35,38)

Polycyclic presentation:

2^9|a^2=h,c^2=e,d^2=egi,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,d^b=def,e^b=ei,f^b=fi,g^b=gi,d^c=de,h^c=hi,e^d=ei,f^d=fi,g^d=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 298805);