#245332

order	512
nilpotency class	3
isoclinism	isoclinic class of #245303
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	#3214	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

921 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)(3,12)(4,18)(5,21)(6,10)(7,11)(8,15)(9,19)(13,17)(14,23)(16,24)(20,25)(22,26)(27,29)(28,31)(30,32)(33,34,38,42)(35,43,45,57)(36,40,48,56)(37,49,52,41)(39,50,53,46)(44,60,61,59)(47,58,51,55)(54,63,64,62);(1,3,7,17)(2,8,11,24)(4,14,20,30)(5,16,6,15)(9,23,26,32)(10,13,21,12)(18,27,25,28)(19,31,22,29)(39,53)(43,57)(54,64)(59,60);(1,2)(3,13)(4,9)(5,10)(6,21)(7,11)(8,15)(12,17)(14,28)(16,24)(18,19)(20,26)(22,25)(23,31)(27,30)(29,32)(33,35)(34,39)(36,44)(37,50)(38,45)(40,54)(41,57)(42,53)(43,49)(46,52)(47,63)(48,61)(51,62)(55,59)(56,64)(58,60);(1,4,7,20)(2,9,11,26)(3,14,17,30)(5,22,6,19)(8,23,24,32)(10,25,21,18)(12,27,13,28)(15,31,16,29)(33,36,38,48)(34,40,42,56)(35,44,45,61)(37,51,52,47)(39,54,53,64)(41,58,49,55)(43,59,57,60)(46,63,50,62)

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=hi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gi,c^b=cg,e^b=eh,g^b=gh,e^c=ei,f^c=fi,g^c=gh,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, 152528);