#419819

order	512
nilpotency class	3
isoclinism	isoclinic class of #419454
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1887 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 32

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72219647, #72219662, #72219665
maximal normal subgroup of	#72219647, #72219662, #72219665

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 327015);