#511894

order	512
nilpotency class	4
isoclinism	isoclinic class of #511893
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1103 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 1103 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:

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

Polycyclic presentation:

2^9|b^2=ei,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fh,h^a=hi,f^c=fh,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	#72250067, #72250084, #72250090
maximal normal subgroup of	#72250067, #72250084, #72250090

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419090);