#451311

order	512
nilpotency class	3
isoclinism	isoclinic class of #420064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2495
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

753 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)(5,6,12,15,11,17,10,16)(7,9,19,13,21,28,20,27)(8,22,25,33,26,32,29,14)(18,31,35,36,23,30,34,24);(5,7,10,20,11,21,12,19)(6,13,16,9,17,27,15,28)(8,23,25,34,26,18,29,35)(14,31,22,36,33,30,32,24);(5,8,10,25,11,26,12,29)(6,14,15,32,17,33,16,22)(7,18,19,34,21,23,20,35)(9,24,27,31,28,36,13,30);(2,4)(5,9,10,27,11,28,12,13)(6,7,15,19,17,21,16,20)(8,24,25,31,26,36,29,30)(14,18,32,34,33,23,22,35)

Polycyclic presentation:

2^9|a^2=efgh,b^2=f,c^2=e,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,h^b=hi,g^c=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, 358507);