#263367

order	512
nilpotency class	3
isoclinism	isoclinic class of #263351
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3497	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

819 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 819 conjugacy classes of subgroups — too many to draw. Only its 144 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) = 128

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

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

Polycyclic presentation:

2^9|b^2=ehi,c^2=i,d^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,f^b=fhi,d^c=dhi,f^c=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306295, #72306307, #72306320
maximal normal subgroup of	#72306295, #72306307, #72306320

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 170563);