#506421

order	512
nilpotency class	4
isoclinism	isoclinic class of #500829
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

580 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 166 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) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=egi,d^2=ei,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^c=gh,h^c=hi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320231, #72322863, #72322876
maximal normal subgroup of	#72320231, #72322863, #72322876
maximal quotient of	#72320842

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413617);