#475529

order	512
nilpotency class	3
isoclinism	isoclinic class of #453966
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1078 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1078 conjugacy classes of subgroups — too many to draw. Only its 234 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:

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

Polycyclic presentation:

2^9|a^2=egi,b^2=ei,c^2=eh,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=di,d^c=de,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 382725);