#251901

order	512
nilpotency class	3
isoclinism	isoclinic class of #251883
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

914 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 914 conjugacy classes of subgroups — too many to draw. Only its 126 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 — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=i,d^2=i,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,f^b=fi,g^b=gh,d^c=di,e^c=eh,f^d=fh

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268106
maximal normal subgroup of	#72268106
maximal quotient of	#72268109

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 159097);