#6463031

order	512
nilpotency class	3
isoclinism	isoclinic class of #6443048
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3153	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2228 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)(3,8)(4,6)(5,7)(9,10,13,18,15,19,14,30)(11,20,23,32,22,33,29,16)(12,24,27,35,26,38,28,17)(21,31,37,36,25,34,39,40);(3,4)(6,8)(9,11)(10,16)(12,25)(13,29)(14,23)(15,22)(17,34)(18,33)(19,32)(20,30)(21,26)(24,36)(27,37)(28,39)(31,35)(38,40);(1,3,5,4)(2,6,7,8)(9,12,14,28,15,26,13,27)(10,17,18,24,19,35,30,38)(11,21,23,37,22,25,29,39)(16,31,33,40,32,34,20,36);(1,4,5,3)(2,6,7,8)(11,22)(16,32)(20,33)(21,25)(23,29)(31,34)(36,40)(37,39);(12,26)(17,35)(21,25)(24,38)(27,28)(31,34)(36,40)(37,39)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6370227);