#94396

order	512
nilpotency class	5
isoclinism	isoclinic class of #7412
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#354
commutator	C₂×C₄²	C₄²
Frattini, cosocle	C₄×#124	C₂²
socle	C₂²	#1393
prime radical	C₂²×D₄	C₄⋊C₄
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#124 ◁ G
LECS	G ▷ C₄×#124 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

217 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 217 conjugacy classes of subgroups — too many to draw. Only its 46 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) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 48 points:

(1,2,4,9,6,10,5,8)(3,11,12,7,14,15,13,16)(17,18,20,25,21,24,34,41)(19,28,30,46,32,39,33,48)(22,26,35,43,36,27,45,44)(23,37,38,29,40,47,42,31);(1,3,5,13,6,14,4,12)(2,7,9,15,10,16,8,11)(17,19)(18,23,26,39)(20,33)(21,32)(22,31,36,29)(24,40,27,28)(25,42,43,46)(30,34)(35,47,45,37)(38,44,48,41)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fi,d^2=fhi,e^2=fh,g^2=i,b^a=bc,c^a=cf,e^a=efgi,g^a=gi,c^b=cg,d^b=di,f^b=fi,g^b=gh,h^b=hi,e^c=ehi,g^e=gi

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, 1592);