#246427

order	512
nilpotency class	3
isoclinism	isoclinic class of #246415
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1246 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 1246 conjugacy classes of subgroups — too many to draw. Only its 178 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) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6)(7,16)(8,12)(9,13)(10,15)(11,17)(14,19)(18,20)(21,22,25,28)(23,29,30,36)(24,32,33,27)(26,34,35,31);(1,2)(3,4)(5,7,10,8)(6,11,15,20)(9,18,19,17)(12,13,16,14)(24,33)(27,32)(29,36)(31,34);(2,4)(5,8,10,7)(6,12,15,16)(9,18,19,17)(11,13,20,14)(21,23)(22,26)(24,34)(25,30)(27,36)(28,35)(29,32)(31,33);(6,13)(9,19)(11,16)(12,20)(14,15)(17,18)(24,33)(27,32)(29,36)(31,34)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 153623);