#401565

order	512
nilpotency class	3
isoclinism	isoclinic class of #394469
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

866 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)(5,6,11,16,10,15,22,30)(7,17,19,9,14,12,28,26)(8,20,24,31,25,32,27,13)(18,34,35,29,21,23,36,33);(5,7,10,14)(6,12,15,17)(8,21)(9,16,26,30)(11,19,22,28)(13,29)(18,25)(20,23)(24,36)(27,35)(31,33)(32,34);(5,8,22,24,10,25,11,27)(6,13,30,20,15,31,16,32)(7,18,28,35,14,21,19,36)(9,23,17,33,26,34,12,29);(2,4)(5,9,22,17,10,26,11,12)(6,14,30,19,15,7,16,28)(8,23,24,33,25,34,27,29)(13,21,20,36,31,18,32,35)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321557
maximal normal subgroup of	#72321557

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 308761);