#138800

order	512
nilpotency class	4
isoclinism	isoclinic class of #138776
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×D₄	C₂³
prime radical	#25754	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#547 ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 631 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) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 24 points:

(1,2)(3,7,8,4)(5,6)(9,10,15,18)(11,14,17,13)(12,19)(16,20)(21,23)(22,24);(1,3,4,6)(2,5,8,7)(9,11,13,10,15,17,14,18)(12,19,21,16,22,24,20,23);(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,24)(18,23)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,d^2=ghi,e^2=gh,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,d^b=dh,e^b=egh,f^b=fhi,g^b=gi,h^b=hi,d^c=dgh,e^c=egh,f^c=fgh,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45996);