#144385

order	512
nilpotency class	4
isoclinism	isoclinic class of #144384
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	C₂³×Q₈	C₂³
prime radical	#23899	C₂
LCS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 841 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)(3,13)(4,10)(5,19)(6,22)(7,12)(8,15)(9,18)(11,24)(14,26)(16,31)(17,25)(20,28)(21,30)(23,29)(27,32);(1,3)(2,8)(4,15)(5,20)(6,23)(7,17)(9,29)(10,25)(11,26)(12,28)(13,19)(14,18)(16,24)(21,32)(22,27)(30,31);(1,4,7,5)(2,9,12,30)(3,14,17,32)(6,21,24,18)(8,25,28,13)(10,11,19,22)(15,16,20,23)(26,29,27,31)

Polycyclic presentation:

2^9|c^2=i,d^2=i,e^2=i,f^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,f^b=fi,g^b=gi,d^c=dg,e^c=ei,f^c=fi,e^d=ei,f^d=fi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51581);