#244623

order	512
nilpotency class	3
isoclinism	isoclinic class of #244583
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#10212	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

955 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 955 conjugacy classes of subgroups — too many to draw. Only its 146 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=h,d^2=h,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,g^b=gi,e^c=ei,f^c=fi,g^c=gi,e^d=eh,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151819);