#395700

order	512
nilpotency class	3
isoclinism	isoclinic class of #393906
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2408
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1891 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302896);