#123045

order	512
nilpotency class	4
isoclinism	isoclinic class of #123031
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1595
commutator, Frattini, cosocle	C₂×C₂²⋊C₈	C₂³
prime radical	#32764	C₂
LCS	G ▷ C₂×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

366 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,7)(4,6)(5,8)(9,16)(10,13)(11,15)(12,14)(17,18,23,29)(19,30,33,45)(20,24,37,41)(21,28,22,27)(25,38,43,32)(26,36,40,34)(31,44,35,42)(39,47,46,48);(2,6)(3,9)(7,15)(8,14)(10,11)(13,16)(18,24)(20,34)(21,22)(25,39)(26,28)(27,40)(29,41)(30,44)(31,47)(32,38)(35,48)(36,37)(42,45)(43,46);(1,3,5,9)(2,7,8,16)(4,11,12,10)(6,13,14,15)(17,19)(18,25)(20,35)(21,38)(22,32)(23,33)(24,39)(26,44)(27,45)(28,30)(29,43)(31,37)(34,48)(36,47)(40,42)(41,46)

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=fh,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fh,g^a=gi,c^b=ci,d^b=df,e^b=egi,f^b=fh,d^c=dg,e^c=eh,e^d=ei

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30241);