#213314

order	512
nilpotency class	3
isoclinism	isoclinic class of #19733
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄×#127
prime radical	C₂³	#532
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

604 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 202 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=fi,d^2=gi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,d^b=dhi,d^c=dh

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, 120510);