#432144

order	512
nilpotency class	3
isoclinism	isoclinic class of #419466
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	#23444	C₂
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1663 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)(7,11,15,19)(8,9)(10,12)(13,20,16,14)(17,18);(5,7)(6,11,12,19)(8,16)(9,14,18,20)(10,15)(13,17);(5,8)(6,9)(7,13,15,16)(10,17)(11,14,19,20)(12,18);(2,4)(5,9)(6,8)(7,14,15,20)(10,18)(11,13,19,16)(12,17)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251367, #72251386, #72251401
maximal normal subgroup of	#72251367, #72251386, #72251401
maximal quotient of	#72251519

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 339340);