#6365648

order	512
nilpotency class	3
isoclinism	isoclinic class of #6365522
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	#3151	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1489 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283124, #72283237, #72283574, …
maximal normal subgroup of	#72283124, #72283237, #72283574, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6272844);