#249753

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	#61935	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 990 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)(19,29)(20,34)(21,28)(22,27)(23,32)(24,36)(25,35)(26,39)(30,45)(31,40)(33,42)(37,44)(38,41)(43,47)(46,48);(5,14)(8,13)(9,16)(12,15)(17,19,22,33)(18,23,27,41)(20,31,36,46)(21,38,39,32)(24,43,34,45)(25,40,44,48)(26,42,28,29)(30,35,47,37);(1,3)(2,7)(4,10)(5,15)(6,11)(8,16)(9,13)(12,14)(17,20)(18,24)(19,30)(21,35)(22,36)(23,40)(25,26)(27,34)(28,44)(29,43)(31,38)(32,46)(33,47)(37,39)(41,48)(42,45);(1,4,6,2)(3,10,11,7)(5,8,14,13)(9,15,16,12)(17,20)(18,25)(19,31)(21,37)(22,36)(23,40)(24,26)(27,44)(28,34)(29,45)(30,38)(32,47)(33,46)(35,39)(41,48)(42,43)

Polycyclic presentation:

2^9|a^2=i,b^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, 156949);