#494017

order	512
nilpotency class	4
isoclinism	isoclinic class of #480314
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
socle	C₂	#20723
prime radical	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₄×C₈ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

750 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)(3,12,15,8)(4,9)(5,10)(6,11)(7,14,20,28)(13,22,26,21)(16,24)(17,23)(18,25)(19,27,31,32)(29,30);(1,3)(2,7,11,20)(4,13)(5,14)(6,15)(8,10,12,25)(9,19,24,31)(16,26)(17,27)(18,28)(21,23,22,30)(29,32);(2,8)(4,16)(7,10)(9,22)(11,12)(13,26)(17,29)(19,30)(20,25)(21,24)(23,31)(27,32);(1,4,5,17,6,16,18,29)(2,9,10,23,11,24,25,30)(3,13,14,27,15,26,28,32)(7,19,12,22,20,31,8,21)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=e,e^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,g^b=gi,d^c=di,g^c=gh,g^f=gi

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, 401213);