#6367361

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
prime radical	#18684	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1510 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:

(1,2,7,12)(3,13,14,15)(4,9,8,16)(5,11,6,10)(17,18,23,27)(19,28,31,40)(20,25,22,36)(21,26,35,43)(24,29,39,32)(30,45,46,42)(33,41,37,44)(34,48,47,38);(1,3)(2,8)(4,12)(5,16)(6,9)(7,14)(10,15)(11,13)(17,19)(18,24)(20,32)(21,34)(22,29)(23,31)(25,40)(26,42)(27,39)(28,36)(30,33)(35,47)(37,46)(38,44)(41,48)(43,45);(1,4)(2,3)(5,15)(6,13)(7,8)(9,10)(11,16)(12,14)(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)(2,9)(3,10)(5,13)(6,15)(7,8)(11,14)(12,16)(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|a^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	#72283007, #72283021, #72283043, …
maximal normal subgroup of	#72283007, #72283021, #72283043, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6274557);