#136602

order	512
nilpotency class	4
isoclinism	isoclinic class of #2238
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#481
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8939
prime radical	#3054	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

415 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 415 conjugacy classes of subgroups — too many to draw. Only its 114 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 — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,d^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,d^b=dg,e^b=ei,f^b=fg,g^b=gi,h^b=hi,d^c=di,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43798);