#140969

order	512
nilpotency class	4
isoclinism	isoclinic class of #125422
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1505
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

420 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(2,4)(5,6)(7,17,21,39)(8,13)(9,28,29,57)(10,15)(11,16)(12,20)(14,18,40,46)(19,42,49,64)(22,43,30,58)(23,52,54,63)(24,35)(25,56,47,67)(26,37)(27,38)(31,48)(32,50)(33,51)(34,53)(36,45,59,66)(41,44)(55,62)(60,68)(61,65);(5,7,18,44,24,22,9,20,11,21,46,50,10,30,29,51)(6,12,28,43,15,41,40,17,16,33,57,58,35,32,14,39)(8,23,45,48,55,49,25,53,27,54,66,68,26,19,47,65)(13,34,56,64,37,31,59,52,38,61,67,42,62,60,36,63);(1,2,3,4)(5,8,24,55,11,27,10,26)(6,13,35,62,16,38,15,37)(7,19,30,54,21,49,22,23)(9,25,46,66,29,47,18,45)(12,31,41,61,33,60,32,34)(14,36,57,67,40,59,28,56)(17,42,58,63,39,64,43,52)(20,48,44,65,51,68,50,53)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=egi,d^2=gh,g^2=i,b^a=bd,c^a=ce,d^a=dg,h^a=hi,c^b=cg,d^b=dh,f^b=fhi,g^b=gi,h^b=hi,f^d=fi

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, 48165);