#472544

order	512
nilpotency class	3
isoclinism	isoclinic class of #469875
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2607
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16824
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

794 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 794 conjugacy classes of subgroups — too many to draw. Only its 176 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) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

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, 379740);