#450286

order	512
nilpotency class	3
isoclinism	isoclinic class of #417541
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2580
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

972 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:

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

Polycyclic presentation:

2^9|a^2=efh,b^2=e,c^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,h^c=hi,e^d=ei,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250537, #72252980, #72256869
maximal normal subgroup of	#72250537, #72252980, #72256869

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 357482);