#401563

order	512
nilpotency class	3
isoclinism	isoclinic class of #394464
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

861 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)(5,6,9,14,11,15,10,24)(7,16,20,26,19,29,17,12)(8,13,22,28,21,27,23,34)(18,30,32,36,33,35,31,25);(7,17,19,20)(8,21)(12,16,26,29)(13,27)(18,32,33,31)(22,23)(25,35,36,30)(28,34);(5,7,9,17,11,19,10,20)(6,12,14,29,15,26,24,16)(8,18,22,31,21,33,23,32)(13,25,28,35,27,36,34,30);(2,4)(5,8,9,22,11,21,10,23)(6,13,14,28,15,27,24,34)(7,18,17,31,19,33,20,32)(12,25,29,35,26,36,16,30)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252634, #72256865, #72256910
maximal normal subgroup of	#72252634, #72256865, #72256910

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 308759);