#213777

order	512
nilpotency class	3
isoclinism	isoclinic class of #21782
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#535	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1199 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=ce,f^b=fh,g^b=gh,f^c=fh,g^c=gh,g^d=gh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 120973);