#357316

order	512
nilpotency class	3
isoclinism	isoclinic class of #23360
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2380
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#13830
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

707 conjugacy classes of subgroups; only the 312 normal subgroups are shown.

normal subgroups only

This group has 707 conjugacy classes of subgroups — too many to draw. Only its 312 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)(3,13,16,8)(4,17,20,27)(5,10)(6,12)(7,11)(9,19,26,21)(14,25)(15,28,22,24)(18,23)(29,31)(30,32);(1,3,6,15,7,16,5,22)(2,8,12,24,11,13,10,28)(4,18,21,30,20,31,19,14)(9,25,27,23,26,32,17,29);(1,4,5,19,7,20,6,21)(2,9,12,27,11,26,10,17)(3,14,15,18,16,30,22,31)(8,23,28,25,13,29,24,32);(2,10,11,12)(8,24,13,28)(9,17,26,27)(23,32,29,25)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ef,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cef,d^b=de,f^b=fi,g^b=gi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264512);