#455196

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

1297 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,d^2=e,e^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,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, 362392);