#250316

order	512
nilpotency class	3
isoclinism	isoclinic class of #250199
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	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

960 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)(18,24)(20,34)(21,22)(25,29)(27,28)(30,44)(31,32)(36,37)(38,42)(39,48)(40,41)(46,47);(2,6)(3,11)(4,13)(7,15)(8,9)(10,16)(17,18,23,28)(19,29,33,40)(20,35,36,26)(21,27,22,24)(25,31,41,32)(30,45,46,43)(34,42,37,38)(39,44,48,47);(17,19,23,33)(18,25,28,41)(20,30,36,46)(21,32,22,31)(24,29,27,40)(26,39,35,48)(34,47,37,44)(38,43,42,45);(1,3,5,11)(2,7,9,16)(4,14,13,12)(6,10,8,15)(17,20,23,36)(18,26,28,35)(19,30,33,46)(21,37,22,34)(24,38,27,42)(25,39,41,48)(29,43,40,45)(31,47,32,44)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157512);