#403012

order	512
nilpotency class	3
isoclinism	isoclinic class of #394465
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1221 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)(5,6,10,23,11,8,21,14)(7,15,19,16,20,12,18,27)(9,13,24,35,25,22,34,28)(17,29,32,30,33,26,31,36);(5,7,10,18,11,20,21,19)(6,12,23,16,8,15,14,27)(9,17,24,31,25,33,34,32)(13,26,35,30,22,29,28,36);(5,8)(6,11)(7,16)(9,22)(10,14)(12,19)(13,25)(15,18)(17,30)(20,27)(21,23)(24,28)(26,32)(29,31)(33,36)(34,35);(2,4)(5,9,10,24,11,25,21,34)(6,13,23,35,8,22,14,28)(7,17,18,31,20,33,19,32)(12,26,16,30,15,29,27,36)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250285, #72257150, #72257201
maximal normal subgroup of	#72250285, #72257150, #72257201

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 310208);