#94388

order	512
nilpotency class	5
isoclinism	isoclinic class of #94311
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1390
commutator	C₂×C₄⋊C₄	C₄²
Frattini, cosocle	C₄×C₄⋊Q₈	C₂²
prime radical	C₂³	#342
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×C₄⋊Q₈ ◁ G
LECS	G ▷ C₄×C₄⋊Q₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

203 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 203 conjugacy classes of subgroups — too many to draw. Only its 44 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) = 64

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1584);