#385801

order	512
nilpotency class	3
isoclinism	isoclinic class of #31414
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

963 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 292997);