#437720

order	512
nilpotency class	3
isoclinism	isoclinic class of #427374
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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

1152 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)(3,4)(5,6)(7,10,14,19)(8,15,17,11)(9,12)(13,20)(16,18);(5,7)(6,10,12,19)(8,16)(9,14)(11,20,15,18)(13,17);(5,8)(6,11)(7,13,14,16)(9,17)(10,18,19,20)(12,15);(2,4)(5,6)(7,10,14,19)(8,11)(9,12)(13,18,16,20)(15,17)

Polycyclic presentation:

2^9|a^2=efg,b^2=f,c^2=ef,d^2=ef,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^c=gi,h^c=hi,e^d=ei,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, 344916);