#93052

order	512
nilpotency class	4
isoclinism	isoclinic class of #93047
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1294
commutator	C₂²×C₈	C₄²
Frattini, cosocle	C₂²×C₂²⋊C₈	C₂²
prime radical	C₂⁵	Q₁₆
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂²×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

382 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 64 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=f,d^2=h,e^2=h,f^2=h,g^2=h,b^a=bc,c^a=cf,e^a=efg,f^a=fh,g^a=gh,c^b=cg,d^b=dh,f^b=fh,g^b=gi,e^c=ehi

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, 248);