#254085

order	512
nilpotency class	3
isoclinism	isoclinic class of #254083
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#36149	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1090 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|b^2=e,c^2=i,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fhi,g^b=gi,e^c=eh,f^c=fi,g^c=ghi,f^d=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261872, #72261882
maximal normal subgroup of	#72261872, #72261882

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 161281);