#476000

order	512
nilpotency class	3
isoclinism	isoclinic class of #459486
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2804
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

714 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 714 conjugacy classes of subgroups — too many to draw. Only its 186 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) = 32

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=f,d^2=h,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^c=de,e^c=ei,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227991
maximal normal subgroup of	#72227991
maximal quotient of	#72268633

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 383196);