#249849

order	512
nilpotency class	3
isoclinism	isoclinic class of #249839
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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1079 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1079 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157045);