#404529

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1022 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)(7,12)(8,21,22,31)(9,15)(10,14)(11,16)(13,23,30,24)(17,32,33,36)(18,27)(19,26)(20,28)(25,34,29,35);(5,7,9,18,11,20,10,19)(6,12,15,27,16,28,14,26)(8,17,24,35,22,33,23,34)(13,29,31,32,30,25,21,36);(5,8)(6,13)(7,17)(9,24)(10,23)(11,22)(12,25)(14,21)(15,31)(16,30)(18,35)(19,34)(20,33)(26,32)(27,36)(28,29);(2,4)(6,14,16,15)(12,26,28,27)(13,21,30,31)(25,32,29,36)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311725);