#112301

order	512
nilpotency class	3
isoclinism	isoclinic class of #97710
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

803 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,d^b=di,e^b=eh,g^b=ghi,d^c=dh,g^c=gi

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, 19497);