#461542

order	512
nilpotency class	3
isoclinism	isoclinic class of #454974
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	#2482
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

954 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)(5,6,10,15,11,16,9,14)(7,12,18,28,19,29,25,33)(8,20,26,13,23,34,24,32)(17,35,36,31,22,30,21,27);(5,7)(6,12)(8,21,23,36)(9,25)(10,18)(11,19)(13,30,32,35)(14,33)(15,28)(16,29)(17,24,22,26)(20,31,34,27);(5,8,9,26,11,23,10,24)(6,13,14,34,16,32,15,20)(7,17,18,21,19,22,25,36)(12,27,28,30,29,31,33,35);(2,4)(5,7)(6,12)(8,22)(9,25)(10,18)(11,19)(13,31)(14,33)(15,28)(16,29)(17,23)(20,35)(21,26)(24,36)(27,32)(30,34)

Polycyclic presentation:

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

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, 368738);