#493895

order	512
nilpotency class	4
isoclinism	isoclinic class of #480314
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂×#1820
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

857 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)(3,12,15,10)(4,16)(5,9)(6,11)(7,14,21,28)(8,13,22,26)(17,25)(18,20)(19,24)(23,27,31,32)(29,30);(1,3)(2,7,11,21)(4,13)(5,14)(6,15)(8,20,22,30)(9,12,24,10)(16,23,25,31)(17,26)(18,27)(19,28)(29,32);(1,4,5,18,6,17,19,29)(2,8,9,23,11,22,24,31)(3,13,14,27,15,26,28,32)(7,20,12,25,21,30,10,16);(4,17)(8,22)(13,26)(16,25)(18,29)(20,30)(23,31)(27,32)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^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, 401091);