#452273

order	512
nilpotency class	3
isoclinism	isoclinic class of #420064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

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

697 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)(5,6,10,14,11,16,9,15)(7,17,20,12,21,29,19,28)(8,22,24,13,26,32,25,31)(18,27,34,33,23,30,35,36);(5,7,10,20,11,21,9,19)(6,12,14,29,16,28,15,17)(8,23,25,34,26,18,24,35)(13,30,22,33,31,27,32,36);(5,8,9,24,11,26,10,25)(6,13,14,22,16,31,15,32)(7,18,20,35,21,23,19,34)(12,27,17,33,28,30,29,36);(2,4)(6,14,16,15)(7,19,21,20)(12,28)(13,22,31,32)(17,29)(18,34,23,35)(27,30)(33,36)

Polycyclic presentation:

2^9|a^2=efgh,b^2=f,c^2=e,d^2=ef,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, 359469);