#6367362

order	512
nilpotency class	3
isoclinism	isoclinic class of #6367198
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1554 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)(3,14)(4,10)(5,12)(6,11)(7,13)(8,15)(9,16)(17,18)(19,28)(20,25)(21,26)(22,36)(23,27)(24,32)(29,39)(30,45)(31,40)(33,41)(34,48)(35,43)(37,44)(38,47)(42,46);(1,3,7,9)(2,8,13,4)(5,10,6,15)(11,16,12,14)(17,19,23,31)(18,24,27,39)(20,32,22,29)(21,34,35,47)(25,40,36,28)(26,42,43,45)(30,37,46,33)(38,41,48,44);(1,4)(2,9)(3,13)(5,16)(6,14)(7,8)(10,11)(12,15)(17,20,23,22)(18,25,27,36)(19,29,31,32)(21,33,35,37)(24,28,39,40)(26,41,43,44)(30,34,46,47)(38,45,48,42);(1,4)(2,10)(3,12)(5,14)(6,16)(7,8)(9,11)(13,15)(17,21,23,35)(18,26,27,43)(19,30,31,46)(20,33,22,37)(24,38,39,48)(25,41,36,44)(28,45,40,42)(29,34,32,47);(18,27)(19,31)(25,36)(26,43)(29,32)(30,46)(34,47)(41,44)

Polycyclic presentation:

2^9|b^2=h,c^2=hi,d^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,f^b=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282994, #72283015, #72283043, …
maximal normal subgroup of	#72282994, #72283015, #72283043, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6274558);