#500775

order	512
nilpotency class	4
isoclinism	isoclinic class of #499849
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×#2199	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1170 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 1170 conjugacy classes of subgroups — too many to draw. Only its 140 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)(7,11)(8,12)(9,14)(10,13)(15,33)(16,25)(17,26)(18,27)(19,32)(20,28)(21,31)(22,29)(23,30)(24,34)(35,36);(1,2,4,3)(5,7,10,16)(6,11,13,25)(8,18,20,35)(9,21,23,17)(12,27,28,36)(14,31,30,26)(15,22,34,19)(24,32,33,29);(5,8)(6,12)(7,15)(9,22)(10,20)(11,24)(13,28)(14,32)(16,34)(17,35)(18,21)(19,23)(25,33)(26,36)(27,31)(29,30);(1,3,4,2)(6,13)(11,25)(12,28)(14,30)(24,33)(26,31)(27,36)(29,32)

Polycyclic presentation:

2^9|b^2=ei,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dei,g^a=ghi,c^b=cg,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, 407971);