#415610

order	512
nilpotency class	3
isoclinism	isoclinic class of #393906
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2580
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1185 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=gh,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,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	#72250020, #72257167, #72257207
maximal normal subgroup of	#72250020, #72257167, #72257207

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322806);