#94327

order	512
nilpotency class	5
isoclinism	isoclinic class of #94311
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1390
commutator	C₂×C₄⋊C₄	C₄²
Frattini, cosocle	C₂×#382	C₂²
prime radical	C₂⁴	#99
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂×#382 ◁ G
LECS	G ▷ C₂×#382 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

305 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 305 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fi,e^2=h,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,e^c=ehi,g^c=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, 1523);