#6367290

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

1525 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)(3,12)(4,9)(5,10)(6,13)(7,11)(8,15)(14,16)(17,18,22,25)(19,23,28,38)(20,30,32,40)(21,33,35,26)(24,34,39,31)(27,41,42,48)(29,45,44,37)(36,46,47,43);(1,3,7,14)(2,8,11,5)(4,15,6,10)(9,16,13,12)(17,19)(18,23)(20,27)(21,29)(22,28)(24,36)(25,38)(26,37)(30,41)(31,43)(32,42)(33,45)(34,46)(35,44)(39,47)(40,48);(1,4,7,6)(2,9,11,13)(3,10,14,15)(5,16,8,12)(17,20,22,32)(18,24,25,39)(19,27,28,42)(21,34,35,31)(23,36,38,47)(26,40,33,30)(29,46,44,43)(37,48,45,41);(1,5)(2,10)(3,13)(4,16)(6,12)(7,8)(9,14)(11,15);(2,11)(3,14)(9,13)(10,15)(18,25)(19,28)(24,39)(26,33)(27,42)(29,44)(30,40)(43,46)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^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	#72306793, #72306794, #72306829
maximal normal subgroup of	#72306793, #72306794, #72306829

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6274486);