#368414

order	512
nilpotency class	3
isoclinism	isoclinic class of #24295
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1290 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=g,d^2=ei,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275610);