#473871

order	512
nilpotency class	3
isoclinism	isoclinic class of #453980
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2782
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

870 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)(5,6,12,20,11,16,10,15)(7,17,26,8,19,27,13,22)(9,14,24,30,25,31,28,32)(18,23,34,36,35,33,29,21);(5,7,12,13,11,19,10,26)(6,8,20,17,16,22,15,27)(9,18,24,29,25,35,28,34)(14,21,30,33,31,36,32,23);(2,4)(5,8)(6,13)(7,15)(9,23)(10,27)(11,22)(12,17)(14,18)(16,26)(19,20)(21,24)(25,33)(28,36)(29,30)(31,35)(32,34);(5,9)(6,14)(7,18)(8,21)(10,28)(11,25)(12,24)(13,29)(15,32)(16,31)(17,33)(19,35)(20,30)(22,36)(23,27)(26,34)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 381067);