#245763

order	512
nilpotency class	3
isoclinism	isoclinic class of #245711
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1068 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,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,e^b=eh,e^c=ei,f^c=fi,e^d=eh,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152959);