#6447743

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437969
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
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3121 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318242, #72318253, #72318324, …
maximal normal subgroup of	#72318242, #72318253, #72318324, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6354939);