#245513

order	512
nilpotency class	3
isoclinism	isoclinic class of #245447
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	#537	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

934 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 152709);