#6368593

order	512
nilpotency class	3
isoclinism	isoclinic class of #6368558
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

1508 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)(3,8)(4,9)(5,7)(6,11)(10,13)(12,15)(14,16)(17,18,24,30)(19,31,20,26)(21,27,38,44)(22,29,23,28)(25,35,41,33)(32,46,34,42)(36,48,47,40)(37,45,39,43);(1,3,4,11)(2,6,9,8)(5,12,13,16)(7,14,10,15)(17,19,24,20)(18,25,30,41)(21,36,38,47)(22,33,23,35)(26,28,31,29)(27,42,44,46)(32,39,34,37)(40,45,48,43);(2,7)(5,13)(6,14)(8,15)(9,10)(12,16)(17,20,24,19)(18,26,30,31)(21,34,38,32)(22,33,23,35)(25,28,41,29)(27,42,44,46)(36,37,47,39)(40,43,48,45);(1,3)(2,8)(4,11)(5,12)(6,9)(7,15)(10,14)(13,16)(17,21)(18,27)(19,32)(20,34)(22,37)(23,39)(24,38)(25,40)(26,42)(28,43)(29,45)(30,44)(31,46)(33,47)(35,36)(41,48);(3,11)(6,8)(12,16)(14,15)(17,22,24,23)(18,28,30,29)(19,33,20,35)(21,37,38,39)(25,31,41,26)(27,43,44,45)(32,47,34,36)(40,46,48,42)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306671, #72306696, #72306707
maximal normal subgroup of	#72306671, #72306696, #72306707

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6275789);