#6368938

order	512
nilpotency class	3
isoclinism	isoclinic class of #6368846
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

1549 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306700, #72306707, #72306722
maximal normal subgroup of	#72306700, #72306707, #72306722

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6276134);