#6516753

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500002
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3370 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,d^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317675
maximal normal subgroup of	#72317675

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6423949);