#405849

order	512
nilpotency class	3
isoclinism	isoclinic class of #395587
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2411
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14919
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

943 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319325, #72319352, #72319460
maximal normal subgroup of	#72319325, #72319352, #72319460

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 313045);