#6463751

order	512
nilpotency class	3
isoclinism	isoclinic class of #6442590
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1897 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320791
maximal normal subgroup of	#72320791

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6370947);