#506501

order	512
nilpotency class	4
isoclinism	isoclinic class of #501332
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

846 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 846 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 20 points:

(5,6,9,13,19,12,17,20,10,14,18,11,8,15,7,16);(1,2,3,4)(5,7,8,18,10,17,19,9)(6,11,12,16,14,13,15,20);(2,4)(5,6)(7,13)(8,12)(9,16)(10,14)(11,17)(15,19)(18,20);(7,17)(9,18)(11,13)(16,20)

Polycyclic presentation:

2^9|a^2=f,b^2=egi,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=deh,e^a=ei,c^b=cg,d^b=di,f^c=fh,h^c=hi,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224852
maximal normal subgroup of	#72224852
maximal quotient of	#72224849

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413697);