#383982

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2688
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14368
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

832 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319622, #72319908, #72320044, …
maximal normal subgroup of	#72319622, #72319908, #72320044, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 291178);