#433700

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

795 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:

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

Polycyclic presentation:

2^9|a^2=fg,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, 340896);