#403111

order	512
nilpotency class	3
isoclinism	isoclinic class of #393934
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	C₂⁵	C₂⁴
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1834 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=ef,d^2=ef,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,h^b=hi,h^c=hi,e^d=ei,f^d=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264449, #72264504, #72264517
maximal normal subgroup of	#72264449, #72264504, #72264517

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 310307);