#455094

order	512
nilpotency class	3
isoclinism	isoclinic class of #454974
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

1504 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:

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

Polycyclic presentation:

2^9|a^2=f,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,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, 362290);