#6366963

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

1506 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(2,7)(4,5)(8,15)(9,13)(10,16)(11,14)(17,18,23,27)(19,28,31,38)(20,25,22,35)(21,26,34,43)(24,36,40,29)(30,45,47,42)(32,41,37,44)(33,48,46,39);(1,2)(3,10)(4,13)(5,7)(6,9)(8,11)(12,15)(14,16)(17,19)(18,24)(20,29)(21,33)(22,36)(23,31)(25,38)(26,42)(27,40)(28,35)(30,32)(34,46)(37,47)(39,44)(41,48)(43,45);(1,2)(3,8)(4,7)(5,13)(6,9)(10,11)(12,16)(14,15)(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);(1,3,6,12)(2,8,9,16)(4,14,5,11)(7,15,13,10)(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|a^2=hi,c^2=hi,d^2=h,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	#72306994, #72307001, #72307019
maximal normal subgroup of	#72306994, #72307001, #72307019

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6274159);