#499103

order	512
nilpotency class	4
isoclinism	isoclinic class of #498890
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1135 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 1135 conjugacy classes of subgroups — too many to draw. Only its 178 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406299);