#245306

order	512
nilpotency class	3
isoclinism	isoclinic class of #245303
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	#10026	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 946 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)(4,6)(5,9)(7,11)(8,14)(12,15)(13,16)(17,18)(19,27)(20,24)(21,33)(22,26)(23,30)(25,36)(28,44)(29,41)(31,42)(32,40)(34,37)(35,39)(38,46)(43,45)(47,48);(2,6)(4,14)(7,10)(8,9)(11,12)(15,16)(23,37)(27,41)(38,48)(43,44);(1,2)(3,7)(4,8)(5,9)(6,14)(10,11)(12,16)(13,15)(17,19)(18,23)(20,28)(21,34)(22,29)(24,38)(25,41)(26,37)(27,33)(30,36)(31,47)(32,45)(35,46)(39,43)(40,48)(42,44);(1,3,5,13)(2,7,9,15)(4,12,14,11)(6,10,8,16)(17,20,22,32)(18,24,26,40)(19,28,29,45)(21,35,36,31)(23,38,37,48)(25,42,33,39)(27,43,41,44)(30,47,34,46)

Polycyclic presentation:

2^9|d^2=hi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gi,c^b=cg,e^b=eh,g^b=gh,e^c=ei,f^c=fi,g^c=gh,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, 152502);