#227162

order	512
nilpotency class	3
isoclinism	isoclinic class of #20286
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	#18494	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

601 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^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=dei,e^b=eh,d^c=dhi,f^c=fi,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 134358);