#6508708

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499892
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
prime radical	#62871	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

2953 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,5,8)(3,6,10,14)(4,7,12,16)(9,13,11,15)(23,34)(35,45)(36,46)(47,48);(1,3)(2,6)(4,11)(5,10)(7,15)(8,14)(9,12)(13,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,35)(28,36)(29,34)(30,37)(31,38)(32,39)(33,40)(41,47)(42,45)(43,48)(44,46);(1,4)(2,7)(3,9)(5,12)(6,13)(8,16)(10,11)(14,15)(17,19)(18,23)(20,27)(21,28)(22,29)(24,35)(25,36)(26,34)(30,41)(31,42)(32,43)(33,44)(37,47)(38,45)(39,48)(40,46);(4,12)(7,16)(9,11)(13,15)(17,20,22,31)(18,24,26,38)(19,27,29,42)(21,32,33,30)(23,35,34,45)(25,39,40,37)(28,43,44,41)(36,48,46,47);(3,10)(6,14)(9,11)(13,15)(17,21,22,33)(18,25,26,40)(19,28,29,44)(20,30,31,32)(23,36,34,46)(24,37,38,39)(27,41,42,43)(35,47,45,48)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72266647
maximal subgroup of	#72266610, #72266647
maximal normal subgroup of	#72266610, #72266647

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6415904);