#507944

order	512
nilpotency class	4
isoclinism	isoclinic class of #499946
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#509	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

504 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=egh,d^2=g,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=gi,h^a=hi,c^b=cg,e^c=ei,f^c=fh,h^c=hi,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239792, #72239808, #72239829
maximal normal subgroup of	#72239792, #72239808, #72239829

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415140);