#142810

order	512
nilpotency class	4
isoclinism	isoclinic class of #142310
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1598
commutator, Frattini, cosocle	C₂²×M₁₆	C₂³
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

672 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 72 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 — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|d^2=gi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,f^a=fhi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=ei,f^b=fg,g^b=gi,d^c=dh,f^c=fg,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250264, #72250581, #72250610
maximal normal subgroup of	#72250264, #72250581, #72250610
maximal quotient of	#72307720

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50006);