#394204

order	512
nilpotency class	3
isoclinism	isoclinic class of #394202
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2522 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(5,6,12,18)(7,19,24,35)(8,25,28,41)(9,15,30,40)(10,17,11,16)(13,23,36,22)(14,31,38,27)(20,45,48,33)(21,44,50,61)(26,54,57,53)(29,42,32,39)(34,52,59,49)(37,55,62,56)(43,51,60,47)(46,58,67,64)(63,65,68,66);(1,2)(3,4)(5,7,11,23,12,24,10,22)(6,13,17,35,18,36,16,19)(8,20,31,51,28,48,27,47)(9,21,32,52,30,50,29,49)(14,33,41,60,38,45,25,43)(15,34,42,61,40,59,39,44)(26,46,56,66,57,67,55,65)(37,58,54,68,62,64,53,63);(2,4)(5,8)(6,14)(7,20)(9,26)(10,27)(11,31)(12,28)(13,33)(15,37)(16,25)(17,41)(18,38)(19,43)(21,46)(22,47)(23,51)(24,48)(29,55)(30,57)(32,56)(34,58)(35,60)(36,45)(39,53)(40,62)(42,54)(44,63)(49,65)(50,67)(52,66)(59,64)(61,68);(5,9)(6,15)(7,21)(8,26)(10,29)(11,32)(12,30)(13,34)(14,37)(16,39)(17,42)(18,40)(19,44)(20,46)(22,49)(23,52)(24,50)(25,53)(27,55)(28,57)(31,56)(33,58)(35,61)(36,59)(38,62)(41,54)(43,63)(45,64)(47,65)(48,67)(51,66)(60,68)

Polycyclic presentation:

2^9|a^2=i,b^2=ei,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,f^c=fi,h^c=hi,f^d=fi

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, 301400);