#245869

order	512
nilpotency class	3
isoclinism	isoclinic class of #245807
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#45269	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1115 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gi,c^b=cg,d^b=dh,e^b=eh,e^c=ei,f^c=fi,e^d=eh,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 153065);