#249834

order	512
nilpotency class	3
isoclinism	isoclinic class of #249799
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	#61797	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 996 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=hi,d^2=h,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,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157030);