#94238

order	512
nilpotency class	5
isoclinism	isoclinic class of #94192
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1391
commutator	C₂²×C₈	C₄²
Frattini, cosocle	#2931	C₂²
socle	C₂	#7237
prime radical	#532	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #2931 ◁ G
LECS	G ▷ #2931 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

207 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 207 conjugacy classes of subgroups — too many to draw. Only its 48 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 64 points:

(1,2,5,11)(3,14,16,44)(4,20,21,39)(6,29,30,59)(7,12,28,41)(8,13,32,43)(9,33,34,50)(10,25,37,22)(15,47,48,38)(17,35,53,61)(18,45,52,60)(19,46,55,63)(23,56,26,58)(24,57,54,64)(27,40,31,42)(36,51,62,49);(1,3,6,17,31,54,22,49,7,18,23,50,8,19,4,15)(2,9,10,14,13,36,29,46,12,35,20,45,42,47,56,64)(5,24,26,48,27,52,21,53,28,55,30,51,32,16,25,33)(11,38,39,63,43,34,58,60,41,62,37,57,40,61,59,44)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fgi,e^2=fghi,g^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fh,g^a=gh,h^a=hi,d^b=dg,d^c=dh,e^d=ei,f^d=fi,g^d=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, 1434);