#93900

order	512
nilpotency class	5
isoclinism	isoclinic class of #7398
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1343
commutator	C₂²×C₈	C₂×C₈
Frattini, cosocle	#1729	C₂²
socle	C₂	#7103
prime radical	C₂²×#135	C₄
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #1729 ◁ G
LECS	G ▷ #1729 ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁

378 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 46 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:

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

Polycyclic presentation:

2^9|a^2=d,c^2=fi,d^2=g,f^2=i,g^2=i,b^a=bc,c^a=ce,e^a=eh,h^a=hi,c^b=cf,d^b=def,f^b=fi,d^c=dh,e^d=ei

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, 1096);