#142808

order	512
nilpotency class	4
isoclinism	isoclinic class of #142258
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1595
commutator, Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂×#2190	C₂
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 643 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)(5,6,12,17)(7,18,22,31)(8,23,9,27)(10,16,11,15)(13,29,33,20)(14,25,34,26)(19,32,24,35)(21,36,28,30);(5,7)(6,13)(8,24)(9,19)(10,28)(11,21)(12,22)(14,30)(15,35)(16,32)(17,33)(18,23)(20,26)(25,29)(27,31)(34,36);(2,4)(7,19)(8,25)(9,26)(10,11)(13,30)(14,27)(15,16)(18,32)(20,28)(21,29)(22,24)(23,34)(31,35)(33,36)

Polycyclic presentation:

2^9|a^2=i,d^2=gi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,f^a=fh,g^a=gi,h^a=hi,c^b=cf,d^b=dg,f^b=fgi,g^b=gi,d^c=dh,f^c=fgi,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	#72250550, #72250581, #72250587
maximal normal subgroup of	#72250550, #72250581, #72250587

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50004);