#455197

order	512
nilpotency class	3
isoclinism	isoclinic class of #454974
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	#60453	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1333 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 362393);