#118127

order	512
nilpotency class	3
isoclinism	isoclinic class of #118124
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

532 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,c^b=cf,e^b=ei,d^c=di,f^c=fi,g^c=ghi

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, 25323);