#250504

order	512
nilpotency class	3
isoclinism	isoclinic class of #250455
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	#447	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

918 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2)(3,11)(4,15)(5,18)(6,10)(7,13)(8,16)(9,21)(12,23)(14,25)(17,26)(19,22)(20,24)(27,28)(29,31)(30,32)(33,34,39,43)(35,44,47,54)(36,49,51,58)(37,52,38,42)(40,48,56,45)(41,53,57,50)(46,55,62,64)(59,60,63,61);(1,3)(2,7)(4,12)(5,19)(6,14)(8,23)(9,25)(10,22)(11,18)(13,21)(15,27)(16,30)(17,29)(20,28)(24,31)(26,32)(33,35,39,47)(34,40,43,56)(36,46,51,62)(37,48,38,45)(41,55,57,64)(42,54,52,44)(49,59,58,63)(50,60,53,61);(7,22)(11,25)(23,31)(27,32)(35,45)(37,38)(40,54)(42,52)(44,56)(46,60)(47,48)(49,58)(50,53)(55,63)(59,64)(61,62);(1,4,6,17)(2,8,10,24)(3,12,14,29)(5,20,21,16)(7,23,22,31)(9,26,18,15)(11,27,25,32)(13,30,19,28)(33,36,39,51)(34,41,43,57)(35,46,47,62)(37,53,38,50)(40,55,56,64)(42,58,52,49)(44,59,54,63)(45,60,48,61)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157700);