#470879

order	512
nilpotency class	3
isoclinism	isoclinic class of #460096
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

864 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,6)(4,8)(5,7)(9,10)(11,19,22,15)(12,16)(13,17)(14,18)(20,29,23,27)(21,34,36,28)(24,32)(25,31)(26,33)(30,38)(35,39,37,40);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,23,25,20)(13,21)(14,22)(16,29,31,27)(17,28)(18,19)(24,37,38,35)(26,36)(30,40,32,39)(33,34);(1,4,5,3)(2,6,7,8)(9,12)(10,16)(11,20)(13,24)(14,25)(15,27)(17,32)(18,31)(19,29)(21,35)(22,23)(26,38)(28,40)(30,33)(34,39)(36,37);(9,13,14,26)(10,17,18,33)(11,21,22,36)(12,24,25,38)(15,28,19,34)(16,30,31,32)(20,35,23,37)(27,39,29,40)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238062
maximal normal subgroup of	#72238062
maximal quotient of	#72237344

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 378075);