#400993

order	512
nilpotency class	3
isoclinism	isoclinic class of #393906
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2535
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1728 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)(7,16,20,17)(8,22,24,13)(9,14)(10,26)(11,15)(12,19,28,21)(18,31)(23,30,25,29)(27,33)(32,34)(35,36);(1,2)(3,4)(5,7,9,21,11,20,10,19)(6,12,14,17,15,28,26,16)(8,18,25,33,24,34,23,35)(13,27,30,32,22,36,29,31);(2,4)(5,6)(7,17)(8,13)(9,14)(10,26)(11,15)(12,19)(16,20)(18,32)(21,28)(22,24)(23,29)(25,30)(27,35)(31,34)(33,36);(5,8,9,25,11,24,10,23)(6,13,14,30,15,22,26,29)(7,18,21,33,20,34,19,35)(12,27,17,32,28,36,16,31)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257124, #72257201, #72257218
maximal normal subgroup of	#72257124, #72257201, #72257218

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 308189);