#144394

order	512
nilpotency class	4
isoclinism	isoclinic class of #144394
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10336
commutator, Frattini, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

927 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 927 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240516, #72338780
maximal normal subgroup of	#72240516, #72338780

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51590);