#250248

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×AGL(1,ℤ/8)	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

987 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)(19,29,33,40)(20,34,37,44)(21,27,22,26)(24,32,41,30)(25,38,43,36)(31,39,35,42)(45,46,48,47);(2,6)(3,11)(4,13)(7,15)(8,9)(10,16)(17,19,23,33)(18,24,28,41)(20,35,37,31)(21,32,22,30)(25,42,43,39)(26,40,27,29)(34,48,44,45)(36,47,38,46);(19,30)(21,22)(24,29)(26,27)(31,46)(32,33)(34,44)(35,47)(36,38)(39,45)(40,41)(42,48);(1,3,5,11)(2,7,9,16)(4,14,13,12)(6,10,8,15)(17,20,23,37)(18,25,28,43)(19,31,33,35)(21,38,22,36)(24,39,41,42)(26,44,27,34)(29,45,40,48)(30,46,32,47)

Polycyclic presentation:

2^9|a^2=h,b^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, 157444);