#114536

order	512
nilpotency class	3
isoclinism	isoclinic class of #100003
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

738 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 68 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 21732);