#6368797

order	512
nilpotency class	3
isoclinism	isoclinic class of #6368766
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	#62405	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1501 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)(3,8)(4,9)(5,7)(6,11)(10,13)(12,15)(14,16)(17,18)(19,27)(20,23)(21,32)(22,33)(24,30)(25,36)(26,39)(28,37)(29,41)(31,43)(34,40)(35,42)(38,44)(45,47)(46,48);(1,3,4,11)(2,6,9,8)(5,12,13,16)(7,14,10,15)(17,19,23,31)(18,24,20,28)(21,34,36,48)(22,37,39,30)(25,41,32,45)(26,43,33,27)(29,38,47,35)(40,44,46,42);(2,7)(5,13)(6,14)(8,15)(9,10)(12,16)(17,20,23,18)(19,28,31,24)(21,32,36,25)(22,33,39,26)(27,30,43,37)(29,46,47,40)(34,45,48,41)(35,44,38,42);(1,3)(2,8)(4,11)(5,12)(6,9)(7,15)(10,14)(13,16)(17,21,23,36)(18,25,20,32)(19,29,31,47)(22,38,39,35)(24,40,28,46)(26,44,33,42)(27,45,43,41)(30,48,37,34);(3,11)(6,8)(12,16)(14,15)(18,20)(24,28)(25,32)(26,33)(27,43)(40,46)(41,45)(42,44)

Polycyclic presentation:

2^9|b^2=hi,c^2=h,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,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

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306654, #72306670, #72306697
maximal normal subgroup of	#72306654, #72306670, #72306697

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6275993);