#250397

order	512
nilpotency class	3
isoclinism	isoclinic class of #250327
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	#62595	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1005 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157593);