#512002

order	512
nilpotency class	4
isoclinism	isoclinic class of #511893
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

947 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 947 conjugacy classes of subgroups — too many to draw. Only its 180 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=eh,b^2=e,e^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fh,h^a=hi,f^c=fhi,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250062, #72250078, #72250090
maximal normal subgroup of	#72250062, #72250078, #72250090

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419198);