#491502

order	512
nilpotency class	4
isoclinism	isoclinic class of #491437
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#33748	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

539 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 539 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=ei,c^2=fh,d^2=g,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,h^a=hi,c^b=ci,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225180, #72225276
maximal normal subgroup of	#72225180, #72225276

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398698);