#359278

order	512
nilpotency class	3
isoclinism	isoclinic class of #23472
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

968 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252681, #72256869, #72256910
maximal normal subgroup of	#72252681, #72256869, #72256910

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266474);