#6368981

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
prime radical	#18713	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1465 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)(3,11)(4,12)(5,8)(6,15)(7,9)(10,16)(13,14)(17,18,23,20)(19,27,30,43)(21,25,36,32)(22,33,39,26)(24,37,28,31)(29,45,47,41)(34,46,48,40)(35,44,38,42);(1,3,7,13)(2,5,9,4)(6,12,16,8)(10,14,15,11)(17,19)(18,24)(20,28)(21,34)(22,37)(23,30)(25,41)(26,43)(27,33)(29,35)(31,39)(32,45)(36,48)(38,47)(40,42)(44,46);(1,4)(2,3)(5,7)(6,14)(8,10)(9,13)(11,16)(12,15)(17,20,23,18)(19,28,30,24)(21,32,36,25)(22,33,39,26)(27,31,43,37)(29,46,47,40)(34,45,48,41)(35,44,38,42);(1,5)(2,8)(3,10)(4,7)(6,14)(9,12)(11,16)(13,15)(17,21,23,36)(18,25,20,32)(19,29,30,47)(22,38,39,35)(24,40,28,46)(26,44,33,42)(27,45,43,41)(31,48,37,34);(2,9)(4,5)(10,15)(11,14)(18,20)(19,30)(25,32)(26,33)(29,47)(31,37)(34,48)(42,44)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=hi,d^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	#72306613, #72306706
maximal normal subgroup of	#72306613, #72306706

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6276177);