#417887

order	512
nilpotency class	3
isoclinism	isoclinic class of #417464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2002 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 2002 conjugacy classes of subgroups — too many to draw. Only its 218 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=ei,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=dei,f^c=fi,h^c=hi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263014, #72263055
maximal normal subgroup of	#72263014, #72263055

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 325083);