#227611

order	512
nilpotency class	3
isoclinism	isoclinic class of #20158
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2358
prime radical	#3482	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

589 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 134807);