#249754

order	512
nilpotency class	3
isoclinism	isoclinic class of #249719
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	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

962 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 962 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156950);