#147518

order	512
nilpotency class	4
isoclinism	isoclinic class of #13229
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1964
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×#41	C₂³
socle	C₂	#10960
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

840 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,h^a=hi,d^b=dh,e^b=ehi,f^b=fhi,h^b=hi,d^c=dhi,e^c=eh,f^c=fh,h^c=hi,f^d=fi,g^d=gi,f^e=fi,g^e=gi,g^f=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279536, #72281046
maximal normal subgroup of	#72279536, #72281046
maximal quotient of	#72279953

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54714);