#500782

order	512
nilpotency class	4
isoclinism	isoclinic class of #499831
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂×#2200	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1163 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(2,3)(5,6,10,14)(7,11,16,25)(8,12,20,29)(9,13,23,32)(15,33,34,24)(17,26,21,30)(18,27,35,36)(19,31,22,28);(1,2,4,3)(5,7,10,16)(6,11,14,25)(8,18,20,35)(9,21,23,17)(12,27,29,36)(13,30,32,26)(15,22,34,19)(24,31,33,28);(5,8)(6,12)(7,15)(9,22)(10,20)(11,24)(13,31)(14,29)(16,34)(17,35)(18,21)(19,23)(25,33)(26,36)(27,30)(28,32);(1,3,4,2)(7,16)(11,25)(15,34)(17,21)(18,35)(24,33)(26,30)(27,36)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407978);