#6366987

order	512
nilpotency class	3
isoclinism	isoclinic class of #6366870
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

1529 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307001, #72307013, #72307019
maximal normal subgroup of	#72307001, #72307013, #72307019

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6274183);