#6368753

order	512
nilpotency class	3
isoclinism	isoclinic class of #6368694
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	#62355	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1503 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)(3,9)(4,6)(5,14)(7,11)(8,15)(10,13)(12,16)(17,18)(19,28)(20,25)(21,38)(22,26)(23,32)(24,37)(27,39)(29,41)(30,47)(31,42)(33,43)(34,40)(35,44)(36,45)(46,48);(1,3,6,12)(2,7,4,13)(5,10,15,11)(8,16,14,9)(17,19)(18,23)(20,34)(21,33)(22,31)(24,41)(25,46)(26,43)(27,42)(28,38)(29,35)(30,36)(32,39)(37,48)(40,45)(44,47);(1,4,6,2)(3,7,12,13)(5,14,15,8)(9,10,16,11)(17,20)(18,24)(19,29)(21,36)(22,35)(23,40)(25,38)(26,45)(27,44)(28,47)(30,32)(31,34)(33,48)(37,39)(41,43)(42,46);(3,10)(5,15)(7,16)(8,14)(9,13)(11,12)(17,20)(18,25)(19,30)(21,37)(22,35)(23,41)(24,38)(26,44)(27,45)(28,47)(29,32)(31,48)(33,34)(36,39)(40,43)(42,46);(2,4)(7,13)(8,14)(9,16)(18,26)(19,31)(20,35)(27,38)(32,33)(36,37)(40,41)(46,47)

Polycyclic presentation:

2^9|b^2=hi,c^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,c^b=ch,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, 6275949);