#495718

order	512
nilpotency class	4
isoclinism	isoclinic class of #33539
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#578	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1022 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=gi,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,h^a=hi,e^b=eh,h^b=hi,d^c=di,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, 402914);