#94809

order	512
nilpotency class	6
isoclinism	isoclinic class of #94807
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7504
commutator	C₄×C₈	C₄²
Frattini, cosocle	C₂×#494	C₂²
prime radical	C₂³	#354
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#494 ◁ G
LECS	G ▷ C₂×#494 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

180 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 36 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=gh,d^2=i,e^2=hi,f^2=hi,g^2=i,b^a=bc,c^a=cf,e^a=eg,f^a=fg,g^a=gh,h^a=hi,c^b=ch,d^b=dfg,f^b=fi,d^c=dgh,e^d=ehi,f^d=fhi,g^d=gi

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2005);