#246995

order	512
nilpotency class	3
isoclinism	isoclinic class of #246987
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×#535	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1129 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1129 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,g^a=gh,c^b=cg,e^b=eh,f^b=fhi,e^c=ei,f^c=fhi,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, 154191);