#7513953

order	512
nilpotency class	3
isoclinism	isoclinic class of #7503371
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×H(3,𝔽₄)
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂×#2782
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1273 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fgh,c^2=f,d^2=fg,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,f^a=fi,g^a=gi,c^b=ch,d^b=df,f^b=fi,h^b=hi,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7421149);