#244427

order	512
nilpotency class	3
isoclinism	isoclinic class of #232749
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2397
prime radical	C₂³	#525
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

547 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=h,d^2=ef,e^2=h,f^2=i,b^a=be,c^a=cf,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,d^c=dh,e^c=ei,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151623);