#248913

order	512
nilpotency class	3
isoclinism	isoclinic class of #248899
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2842 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 20 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72339605, #72339606

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156109);