#144396

order	512
nilpotency class	4
isoclinism	isoclinic class of #144396
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

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

normal subgroups only

This group has 919 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)(4,16)(5,8)(6,11)(7,13)(9,19)(10,20)(14,28)(15,25)(17,23)(18,24)(21,26)(22,32)(27,30)(29,31);(1,3)(2,7)(4,17)(5,14)(6,15)(8,22)(9,25)(10,27)(11,23)(12,16)(13,19)(18,32)(20,31)(21,30)(24,29)(26,28);(2,8)(5,20)(7,22)(9,26)(10,11)(12,28)(13,17)(14,31)(16,24)(18,21)(23,27)(25,29)

Polycyclic presentation:

2^9|d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fgh,g^a=gi,h^a=hi,c^b=cf,d^b=di,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	#72240517, #72286146
maximal normal subgroup of	#72240517, #72286146

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51592);