#432885

order	512
nilpotency class	3
isoclinism	isoclinic class of #427374
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2760
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1236 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 340081);