#245886

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1009 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)(3,10,11,15)(4,6,13,8)(7,14,16,12)(17,18,23,28)(19,29,31,39)(20,33,36,42)(21,26,22,27)(24,37,40,32)(25,34,43,35)(30,38,45,41)(44,47,48,46);(2,6)(3,11)(4,13)(7,15)(8,9)(10,16)(17,19)(18,24)(20,30)(21,37)(22,32)(23,31)(25,41)(26,29)(27,39)(28,40)(33,44)(34,47)(35,46)(36,45)(38,43)(42,48);(17,20,23,36)(18,25,28,43)(19,30,31,45)(21,35,22,34)(24,38,40,41)(26,42,27,33)(29,44,39,48)(32,47,37,46);(1,3,5,11)(2,7,9,16)(4,14,13,12)(6,10,8,15)(18,26)(19,31)(21,22)(24,39)(25,42)(27,28)(29,40)(30,45)(33,43)(34,35)(38,48)(41,44)

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=hi,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, 153082);